Physics

ELECTROMAGNETISM The study of electrical and magnetism phenomena and the relationship between them.

History of Electromagnetism Michael Faraday & James Clerk Maxwell They were further developed But, Maxwell expressed Faraday’s idea as mathematical equations alongside his own. Hans Christian Oested Found-out that there is connection between electricity and magnetism when he noticed that an electric current deflects a compass needle.

Rubbing a piece of amber would enable it to attract bits of straw. “ This phenomenon is already known in ancient Greece.” The world “Electron”, the negatively charged subatomic particles, came from the Greek word which means “Amber”. History

Electric Charge Like mass, is a fundamental property of certain of the elementary particles of which all matter is composed. Fundamental physical quantity responsible for electric phenomena.

Two kind of Electric charge Positively charge Negatively charge Fundamental rule of all electric phenomena: Like charges it repels. Opposite charges it attracts. “ This phenomena was discovered by Charles Du Fay in 1733. The basic law of Electrostatics.

+ + + + + + + + + + + + + + + + + + F F F Repulsion Attraction F

Charged Particles Matter is made-up of atoms. Each atom consist of Proton, Electron and Neutron. Proton are particles with positive charge. Electron are particles with negative charge. Neutron are particles with no charge. “ Protons and Neutron are bound to form the nucleus of the atom. Electron are mobile, moving around the nucleus and can move from one nucleus to another.”

Quantization of charge Each Proton has a charge of +1.6x10 -19 C Each Electron has a charge of -1.6x10 -19 C The Unit of Charge is coulomb (C). The charge 1.6x10 -19 C is denoted as e. Hence, each proton has charge of +e and each electron has charge of –e. Since the charge of all object ultimately depends on the number of Protons and Electrons, all charges are discrete values of e. There can only be 1e, 2e, 3e . . . . n e, but, it is not possible to have 0.5e or 0.125e.

Q = n e Where: Q = Charge (Coulomb) n = “quantity” Integer ( a complete entity or any of the natural number with + & -) e = (1.6 x 10 -19 C)

Sample problem: How many electrons must be added to a neutrally charged body to give a net charge of -1C? An object has a net positive charge of 0.08 C if 1.5 x10 18 electron were transferred to that object, How much is the new charge?

CHARGING There are different ways of making an object positively and negatively charged.

Charging by Friction It happens when one object rubbing with another object allows the transfer of electrons from one to the other.

Charging by Contact It happens when electrons are transferred by simply touching one object with another.

Charging by Induction It allows the movement of charges within a conductor even without touching or rubbing it with another object.

Electricity conduction Most substances conduct electricity either very well or very badly.

Conductors Insulators Semi-conductors Super-conductors Every materials can be classified accordingly:

Conductors Through which charge can flow easily from one substance to another. Has high electron mobility. (free electron flow) Insulators Materials that do not allow much movement of charge Has low electron mobility. (few or no free electron flow)

Semiconductor Materials that has varying conducting properties depending on the impurities and the charges present on the material. Superconductor Materials that become perfect conductors at extremely low temperature.

State that the Force between two charges is proportional to the product of the charges and is inversely proportional to the square of the distance between them. COULOMB’ S LAW

Gave a quantitative description of the strength of attraction and repulsion between charges. q 1 and q 2 are the amount of charges in the particles (in coulombs or C). r is the distance between two charged particles (in meter). k is the proportionality constant = 9 x10 9 Nm 2 /c 2 . ε 0 is a permeability constant = 8.854 x10 -12 c 2 /Nm 2 .

Sample Problem: A hydrogen atom is composed of an electron and a proton. The Bhor radius of the hydrogen atom is 5.30 x10 -11 m. Compute for the electrical force between the proton and the electron in the atom.

Electrostatic Force (F) - + F r Q 1 Q 2 Electrical & Static Force

Notation of Electrostatic Force (Like charge repel, unlike charge attract.) + - + - - + r r r F F F F F F

Sample problem: Three point charges in a plane forming a right triangle, as shown figure below. Find the magnitude of electrostatic force acting on each charge. + + - 0.2 m 0.3 m Q 1 = 1.0 nC Q 2 = 3.0 nC Q 3 = 2.0 nC

Find the electrostatic force between two electrons 2 mm apart? Two identical charge of 4 µC each are 10 mm apart. Find the Electrostatic force. Two point charges Q 1 = 4 µC, Q 2 = 2µC are 30 cm apart. Three point charges are along the same line, as shown in the figure below. Find the Electrostatic force between each charge. Problem solving: Q 1 = +3µC Q 1 = -5µC Q 1 = +8µC 20 mm 35 mm

Force at a Distance Forces that one object can exert to another object with or without physical contact between the objects. Examples are Gravity and Electrostatic force . Object with mass surrounded by gravitational field . Object with charge are surrounded by electric field .

Law of Gravitation Gravitational field

Equations: The equation of gravity: Universal Law of Gravitation The equation of Electrostatic force: Coulomb’s Law

Electric Field Field of force that surround a charged object or particle. Force per unit Charge. It is denoted as E, and its unit is Newton per coulomb:

Where: E = Electric field Q = Charge r = radius of the field k = proportionality constant Electric field equation

Made of infinitely many electric field vectors. Electric of a positive charges directed away from the charge. The electric field of negative charge is directed toward the charge. +Q -Q P P

Drawing Electric Field Lines The lines must begin on positive charges (or infinity) The lines must end on negative charge (or infinity) The number of lines leaving a positive charge (or approaching a negative charge) is proportional to the magnitude of the charge.

Sample Problem: 1. Find (a) the magnitude of an electrons electric field at 50.0 cm away from the electron. (b) if the another electron is placed at this distance, what would be the magnitude of electrostatic force between the electrons? (c) Is the force attractive or repulsive? 2. Two charges, Q 1 = +1.5 x 10 -8 C and Q 2 = +3.0 x 10 -8 C are 100 mm apart. What is the magnitude of the electric field halfway between them?

GAUSS’ LAW A relation between the electric field at all the points on a closed surface and the total charge enclosed within the surface. Named after Karl Friedrich Gauss (1777-1855 An alternative to Coulomb’s Law. Uses symmetry properties of given charges to simplify electric field calculations.

The strength of an electric field over an area in field region. Quantifies the notion “number of field lines crossing a surface”. The dot product of electric field vector passing through the area and the area vector. Where: E = Electric field A = Area of the surface θ = Angle of elevation of the surface Electric Flux ( Φ )

GAUSS’ LAW Φ = EA cos 0 0 E A

Sample problem: If the electric field in the region has a magnitude of 2.0 x 10 3 N/C and passing through the surface with an area 0.0214 m 2 . The area vector is oriented at an angle of 50 0 with respect to the electric field. Find the electric flux.

Gaussian surface A hypothetical surface immersed in electric field. It may or may not enclose a charge. Can be of any shape you wish to make it, but the most useful surface is one that mimics the symmetry of the problem at hand. The area vector is always the face outside the enclosed surface.

Net flux Gauss’ Law in mathematical form: Where Σ q enc is the total charge enclosed. Any Gaussian surface that does not enclose any charge has zero electric flux. If a Gaussian surface encloses a positive charge (or positive sum of several enclosed charges), the electric flux is positive. If a Gaussian surface encloses a negative charge (or negative sum of several enclosed charges), the electric flux is negative.

No enclosed charge (Zero flux) Positive charge enclosed (Positive flux) Negative charge enclosed (Negative flux)

Sample problem: Given five charges of values Q 1 = Q 5 = +3.1 nC, Q 2 = Q 4 = -5.9 nC & q 3 = +3.1 nC, find the net electric flux through the Gaussian surface S shown in the figure below. - - - + + Q 1 Q 2 Q 3 Q 4 Q 5

ELECTRIC POTENTIAL ENERGY Potential energy due to the location of a charge in an external electric field. If a charge Q 0 is within the electric field of another charge Q , the potential of Q 0 is Where r is the distance of Q 0 from Q .

Work done in Electric Field It is negative change in potential energy. If the charge moves between two points with different electric field intensity.

Conclusion: When a positive/negative charge moves in the direction of an electric field, the field does positive/negative work and the potential energy decreases. When positive charge moves in the direction opposite to an electric field, the field does negative work and the potential energy increases.

ELECTRIC POTENTIAL Electric potential energy per unit charge arises due to location within an electric field. Related to electrostatic potential energy in the same way electrostatic force is related to electric field.

For a point charge Q , the electric potential at point r away from the point charge is: If a test charge Q 0 is placed in a region where the electric potential is V , the electric potential energy of the point charge is

The unit is volts (v) 1V = 1 J/C Also known as voltage .

SAMPLE PROBLEM: What is the electric potential at a distance of 5.29 x 10 -11 m from the proton? What is the potential energy of the electron and proton at this separation?

Electric Potential and Electric Field The relationship between electric potential V and electric field E is Where r is the distance from the charge of the point under consideration.

SAMPLE PROBLEM: (a) Find the magnitude of a proton’s electric field at 50.0 cm away from it. (b) What is the electric potential at this distance?

POTENTIAL DIFFERENCE Work done per unit charge as a charge is moved between two points in an electric field. If a test charge Q is moved from point A to point B , the potential difference between A and B is:

Potential of the earth arbitrarily said to be zero. Ultimate responsible for the movement of charge and generation of electric current.

Potential Difference ( ∆V) Can be either positive or negative with respect to the earth, depending on the nature of the charge. W AB is positive the electric potential at B will be higher than the electric potential at point A . W AB is negative the electric potential at B will be lower than the electric potential at point A . W AB is zero the electric potential at B will be the same as the electric potential at point A .

SAMPLE PROBLEM: The work done on a 5.0 C charge is 7.5 J as it is moved from point A, where the potential difference is 2.0 V, to another point B. What is the electric potential difference between points A and points B? What is the potential at point B?

POTENTIAL FOR MULTIPLE CHARGES Calculate the separate potentials of each charge. Add the potentials with these signs corresponding to the sign of the charge. Where n is the number of charges.

SAMPLE PROBLEM: The three charges in with Q 1 = 8x10 -9 C, Q 2 = 2.0x10 -9 C and Q 3 = -4.0x10 -9 C; are separated by distance r 21 = 0.03m and r 31 = 0.05m. Find: Potential due to Q 2 at the point occupied by Q 1 . Potential due to Q 3 at the point occupied by Q 1 . Net potential at point occupied by Q 1 .

Also called condenser A device to stores charge in the electric field between its plates. The plates carry charges of the same magnitude and opposite sign. Ant two parallel conductors separated by an insulator (or vacuum). Symbol:

CAPACITANCE The ability of a capacitor is to store energy. The ratio of charge to potential difference. The unit is Farad (F) = coulomb/volt Capacitance depends on: Area of the plates Distance between the plates Nature of insulating materials (Dielectric)

Space between the plates has uniform electric field. The potential difference ( V ) voltage between the plates a and b is given by: Where E is the electric field and r is between plates

Capacitance for parallel plates capacitors The capacitance of a parallel plate capacitor: Where: ε 0 = permeability constant of the free space = 8.85 x 10 -12 c 2 /Nm 2 = 8.85 x 10 -12 F/m A = area of the plate r = distance between the plate

A larger area will have less repulsion between charges. A greater separation means lesser charge is drawn and the capacitance is less. + Q - Q A r

Sample problem: The plates of a parallel-plate capacitor are 5.00 mm apart and 2.00 m 2 in area. A potential difference of 10,000 volts is applied across the capacitor. Compute the capacitance, the charge on the plate and the magnitude of the electric field in the space between them.

DIELECTRIC Insulating/non-conducting material between the plates of the capacitor. Its function include: Solve the mechanical problem of maintaining two plates at a very small separation w/o actual contact. Increased the maximum possible voltage between the two plates w/o experiencing a “dielectric breakdown”. Dielectric breakdown happens when the dielectric materials becomes slightly conducting. Increase capacitance.

Dielectric Constant Ratio between the capacitance of a capacitor when a dielectric material is present ( C ) and its capacitance when the space between its plate is a vacuum ( C 0 ) Where: k = dielectric constant C = Capacitance if there is dielectric C 0 = capacitance without dielectric

Ratio of the permittivity of the dielectric and the permittivity in a vacuum. Always greater the 1 because C>C 0 , when the charge on the plate is constant. It is also unitless. Where: k = dielectric constant ε = Capacitance if there is dielectric ε 0 = capacitance without dielectric = 8.85 x 10 -12 C 2 /Nm 2

Dielectric Materials PLATE PLATE + -

Sample problem: The parallel plates of a capacitor have an area of 2.00 x 10 -1 m 2 and have a separation distance of 1.00 x 10 -2 m and are connected to 3000 volts power supply. The capacitor is then disconnected from the supply, and an dielectric is inserted between the plates, Find that the potential difference decreases to 1000 volts while the charge on each plate remain constant. Find the following: a) Original capacitance (C), b) magnitude of charge on each plate, c) capacitance, C after dielectric is inserted, d) the dielectric constant k of the dielectric e) permittivity of the dielectric

EQUIVALENT CAPACITANCE Capacitance of the single capacitor that can replace a set of interconnected capacitors. C T V T C 1 C 3 C 2 V T

Capacitors in Series Connection The end of the capacitor is connected to the end of the adjacent capacitor. C 1 C 3 C 2 V T

The relationship of individual capacitances, charges and voltage to equivalent capacitance, charge and voltage respectively are as follow:

Capacitors in Parallel connection Each has one end joined to the corresponding end of all the other capacitors. C 1 C 2 C 3 V T

SAMPLE PROBLEM: Two capacitors one is 491 µF and the other is 30 µF are connected in series across a 12 volts battery. Find the equivalent capacitance of the combination, the charge on each capacitor and the potential difference across it. Two capacitor one is 5 F and the other 2 F are connected in parallel across a 100 volts battery. Find the equivalent capacitance of the combination, the charge of each and the potential difference on each capacitor.

What is the equivalent capacitance of the mixed series and parallel capacitors shown below? 5mF 4mF 18mF

ELECTRIC CURRENT A flow of charge from one place to another. The unit is Ampere , which equal to a flow of 1 coulomb per second.

Moving charges as a current Its described as a stream of moving charges. May range very small currents such as the nerve impulses to a large as the solar wind emitted by the sun. There must be a “net” flow of charges towards one direction.

When moving charges is not a current When there is no net flow of charge even though there are actual movement. Example: Electrons of a copper conductor in absence of electric potential. Electrons just move randomly the charge flowing charge flowing to one direction is equal to those flowing to the other direction.

Electric current in a conductor An isolated conductor in absence of electric potential contains free flowing electrons but no electric current. Isolated conductor charges

A conductor connected to a dry cell or battery has the necessary electric potential to influence the flow of charges towards one direction, hence producing current. Battery + - Conductor Charges Direction of charges

Electric current ( I ) is defined as the amount of charges passing through a hypothetical plane intersecting the conductor per unit of time. Its unit is coulomb per second (C/s), also called ampere (A). Where: I = Current (ampere, A) Q = Charge (coulomb, c) t = Time (second, s)

- - - - - - t = t 0 t = t 0 + 1 s plane plane

Independent to the selection of hypothetical plane a b c a’ b’ c’ I I

Sample problem: A wire carries a current of 0.8 ampere. How many electrons passes every section of the wire every one second?

Current is a scalar quantity Electric current is moving along a conductor has only two possible directions. Electric current are scalars. Adding and Subtracting the current does not consider the orientation of the conductor in space.

DIRECTION OF CURRENT In reality, electric current are movement of electrons along the conductor. For historical reason, current is treated as flow of positive charges to the direction opposite that of the actual movement of electrons.

These positive charges are not actual particles. They are called holes , vacant spaces where there should be an electron. The charge of a hole is +1.6 x 10 -19 C . Electrons are known as negative charge carriers . Holes are known as positive charge carriers .

Drift Speed The net motion of charged particles as a group: Where: I = electric current (A) n = charge concentration v d = drift velocity (m/s) e = charge of electron A = cross-sectional area of conductor(m 2 ) Usually very small (10 -5 or 10 -4 m/s) compared to random motion of charges (10 -6 m/s) I in I in A

Current Density Current per unit of cross-sectional area of a conductor. A vector quantity with the same direction as the motion of positive charge carriers. Where: I = electric current (A) J = current density (A/m 2 ) n = charge concentration v d = drift velocity (m/s) e = charge of electron A = cross-sectional area of conductor(m 2 )

Sample Problem: A 491 gauge copper wire has a nominal diameter of 0.64 mm. This wire carries a constant current of 1.67 A to a 4,910 watts lamp. The density of free electron is 8.5 x 10 28 electrons/m 3 . Find the current density and the magnitude of drift velocity.

Types of Current Direct current The direction of current is constant. The graph of current vs time is a straight line. Developed by Tomas Alva Edison Soon replaced by alternating current as primary means of transmitting electricity, but still used in battery operated devices.

Alternating Current The direction and magnitude of the current continuously changes between two extremes. The graph of current vs time is sinosoid. Developed by Nikola Tesla and George Westinghouse , forming rivalry with Thomas Edison on War of the Currents . The most commonly used method of electric transmission today.

Direct Current Alternating Current I (A) t (s) I (A) t (s)

Electric Resistance Property of the conducting medium that weakens the transmission of electric current. Denoted as R and its unit is Ohm ( Ω ) . Where: R = Resistance (Ohm, Ω ) ρ = resistivity ( Ω m) L = Length of the wire (m) A = cross-sectional area of a wire(m 2 )

0.0038 1.6 x 10-8 Silver 0.0036 11 x 10-8 Platinum 0.00088 98 x 10-8 Mercury 0.0043 21 x 10-8 Lead 0.005 12 x 10-8 Iron 0.0039 1.7 x 10-8 Copper 0.0039 2.6 x 10-8 Aluminum α (k -1 ) ρ (Ω.m) Substance and their temperature coefficient. Approximate resistivities (at 20 0 C)

Sample Problem: A piece of 1.0 m wire has a resistance of 0.19 ohms. Calculate the resistivity of the wire. The cross-sectional area of the wire is 0.5 mm 2 . ρ L A

Resistivity & Conductivity Resistivity ( ρ ) Measure of how much resistance a material possesses against electric current. Intrinsic property of a material that depends on its electronic structure. Conducting material Electric field

Measure by placing the material between two plates with constant electric field ( E ) and taking the ratio of electric field and current density ( J ) . Varies with temperature Where: ρ = resistivity ( Ω m) E = electric field (N/c) J = current density (A/m 2 )

Conductivity Measure of how the material is capable of conducting electricity. Reciprocal of resistivity.

Variation of Resistivity with Temperature Over a wide range of temperature, the graph of resistivity vs temperature of metal is linear. 400 200 0 1200 1400 2 8 0 4 6 10 600 800 1000 Resistivity 10 -8 Ω m Room temperature Temperature (Kelvin)

Variation of Resistivity with Temperature Thus it can be represented by a Linear equation. Where: ρ = resistivity ( Ω m) ρ 0 = resistivity at room temperature ( Ω m) T = temperature (Kelvin,K) T 0 = room temperature (K) α = coefficient of resistivity (K -1 )

The Temperature coefficient of resistivity ( α ) determines how much resistivity increases with temperature. Its unit is (per Kelvin)K -1 .

Sample Problem: What is the resistivity of iron at 200K? Use the values of resistivity (at room temperature) and temperature coefficient of the resistivity in the handout.

Ohm’s Law The current I (Ampere, A) is directly proportional to the potential difference V (Volt,V) with resistance R (ohms, Ω ) as the proportionality constant.

Assumed that the resistance does not vary with voltage or current. Not all conducting material follow “Ohm’s Law”. Those are follow are said to be ohmic , while those that do not are said to be non ohmic .

Current Potential Difference graph of a 1000 W resistor , an Ohmic device. -4 -2 0 +2 +4 -2 +2 0 Current (mA) Potential Difference (V)

Current vs Potential Difference graph of a pn junction diode , a non-ohmic device. -4 -2 0 +2 +4 -2 +2 0 Current (mA) Potential Difference (V)

Single Loop Circuit Circuit Close network of electronic devices through which current constantly flows. EMF Device Maintain potential difference. Provides steady flow of charge. EMF stand for Electromotive force . R EMF I + - + - I

The Resistor Provides a resistance to the circuit. It was specially designed to only provide certain amount of resistance. An Ohmic device Such conductor device. It was verified experimentally by the German physicist Georg Ohm (1787-1854).

Electromotive Force A circuit consists of electrons from the negative terminal of a battery to the positive terminal of the battery. Electrons must then return to the negative terminal, or current will stop flowing. The electron are forced into this higher potential by a electromotive force. EMF

EMF Devices: Battery or Dry Cell Electrochemical Cell Electric Generator Photovoltaic Cell

Internal Resistance The resistance found inside real batteries Lessen the output voltage of the battery. Denoted as r i Its SI unit is Ohms (Ω) . A real battery is now drawn as: EMF r i

Terminal Potential Difference ( TPD ) The output voltage of a source of emf after internal resistance takes effect. The equation used to solve for terminal potential difference is: TPD = E – Ir i Where: TPD = voltage across the source (V) E = voltage if the source is ideal emf (V) r i = internal resistance of the source (Ω) I = current flowing through the battery (A)

Sample Problem: A 6.0 V battery is connected to an external 6.0 0hms resistor. What is the value of the current flowing through the external circuit if there is no internal resistance, What is the value of the current flowing through the external circuit when the internal resistance is 0.3 ohms?

Resistors in Single Loop Circuit

Where: R is resistance, I is electric current and V is electric potential difference. Resistors in Series Circuit. R 3 V T I T + - + R 2 + R 1 + - - - R T

Equivalent resistance in a Series Circuit

Sample problem: Resistors R 1 = 2.00 ohms, R 2 = 3.00 ohms and R 3 = 4.00 ohms are in series connection with a voltage source of 100.0 volts. Find the equivalent resistance, electric current and electric potential difference.

Resistor in Parallel Circuit R 3 V T I T + - + R 2 + R 1 + - - - R T I 3 I 2 I 1

Equivalent resistance in a Parallel Circuit

Sample problem: Resistors R 1 = 3.00 ohms, R 2 = 5.00 ohms and R 3 = 7.00 ohms are in parallel connection with a voltage source of 110.0 volts. Find the equivalent resistance, electric current and electric potential difference.

Resistors in Single Loop Circuit Resistor in Series-Parallel Circuit R 3 V T I T + - + R 2 + R 1 + - - - R T

The Power in the Circuits Flow of current across a circuit.

Movement of a charge across a electric device: It moves from higher potential to lower potential. Hence, there is a decrease in potential energy. Q

If there is a decrease in potential energy, there must be a transmission to another form of energy. Light bulb: to heat and light. Electric motor: to mechanical energy Resistor: Internal energy/heat.

The rate at which electric potential energy is transformed to another form of energy is the POWER in the circuit.

Sample Problem: A current flowing through a 25.0 ohm resistor is 2.0 A. How much power is dissipated by the resistor.

MULTILOOP CIRCUIT Provides multiple paths for current. When one component was cut-off, others can still function.

What happen when one component in a series circuit was cut-off?

What happen when one component in a multiloop circuit was cut-off?

Current in a Multiloop Circuit The point where three or more segments of the conductor meet is called the junction. The current split at the junction. Junction current

GUSTAV KIRCHHOFF German physicist who, in the collaboration with Robert William Bunsen, develop ed the science of spectrum analysis. He showed that each element, when heated to incandescence. He produced a characteristic pattern of emission lines. He formulated Kirchhoff’s Law for electric circuit. (1824-1887)

In any closed circuit, the algebraic sum of all EMF’s and potential drop is equal to zero. (Using loop direction) KIRCHHOFF’S LAW R 2 + Emf 1 + - R 1 + Emf 2 + - R 3 + Loop 1 Loop 2 I 1 I 2 I 3 -

At any point in a circuit, the sum of the currents leaving the junction point is equal to the sum of all the current entering the junction point. (Using current direction). R 2 + ε 1 + - R 1 + ε 2 + - R 3 Junction point I 1 I 3 I 2 +

Sample Problem: In a given circuit below, Find: a) I 1 , b) I 2 and c) I 3 10 Ω + 9v + - 15 Ω + 12v + - 5 Ω I 1 I 3 I 2 +

RC CIRCUIT (Resistor and Capacitor in a circuit)

Resistor- Capacitor in a circuit. R + - C S 1 S 2 ε + - Where: ε = Batteries (Emf) S 1 & S 2 = Switches R = Resistor C = Capacitor Open Close

Charging a capacitor R + - C S 1 S 2 ε + - I I I I I closed open Where: V R = Potential difference across the resistor. V C = Potential difference across the capacitor. I

Current I O at the moment S 1 closed ( t = 0) Current I at any time t after S 1 closed: After some time t The charge of the capacitor (q) increases Current ( I ) decreases.

Until the capacitor reaches its equilibrium charge (q eq ), happen when V C reaches V C = ε , which result to I = 0

Charge and current of the capacitor at any given time t after t = 0.

The time constant ( τ ) of RC series circuit. The unit of time constant is second. At time t = τ Q = 0.63 C ε I = 0.37 I o The charging time of RC circuits are often stated in terms of time constant.

Sample Problem: A resistor with resistance R=1.0 x 10 6 Ω , capacitor with capacitance C=2.2 x 10 -6 F, a voltage source with ε = 100 v, and a switch are all connected in a single loop series circuit. The switch is initially open. When the switch is closed, calculate: Initial current across the resistor Equilibrium charge of the capacitor Time constant of the circuit Current through the resistor after 5 seconds Charge of the capacitor after 5 second Charge of the capacitor at t = τ

Introduction to Magnetism The first known magnet are the stoned exposed to earth’s magnetic field called Loadstones , discovered by early Greeks and Chinese. Magnet are surrounded by a field of force called magnetic field . The magnetic force, force exerted by magnets to magnetic materials is a force at distance, just like gravity and electric force. One of the earliest applications of magnetism is the magnetic compass. Diskette, ATM cards and some other storage device contain tiny bits of magnetic materials. Exposure to magnetic field would damage these devices.

Magnetic field of magnets The magnetic of a magnet has the greatest concentration on its two ends called poles . Magnetic field line are drawn to be emanating from the north pole and terminates to the south pole . N S N S N S Bar Magnet Horseshoe Magnet C-shaped Magnet

The magnetic field lines is made up of infinite number of magnetic field vectors. Magnetic field vectors are drawn tangent to the magnetic field line. This method of visualizing magnetic fields was proposed by Michael Faraday , who initially called magnetic field lines of force or line of induction .

N S Magnetic field Magnetic field

Rules in drawing magnetic field lines The direction of the tangent to a magnetic field line at any point gives the direction of the magnetic field vector at that point. The spacing between the lines represents the magnitude of the magnetic field. Magnetic field lines emanate from the north pole and terminate at the south pole.

Polarity of Magnet Similar poles repel and opposite pole attracts. When a magnet is divided into two (or several parts), each part has its own north and south poles. As far as the current theories of magnetism are concerned, there are no magnetic monopoles.

N N N S S S N N N S S S REPULSION REPULSION ATTRACTION

Definition of Magnetic Field Magnetic field is defined in terms of force it can exert on a charge particle, called magnetic force. Magnetic force is a cross-product: Where: F = Magnetic force (Newton) q = charge (coulomb) v = velocity (m/s) β = Magnetic field (Tesla)

The magnitude of the magnetic force is: The direction of the magnetic force can be determined using the right-hand rule . Where: θ is the angle between velocity and magnetic field.

Right-hand-rule: Long, straight Current: Grasp the wire with your right hand so that your thumb point in the direction of the current. The curled fingers of that hand point the direction of the magnetic field. Current loop: Grasp the loop so that the curled fingers of your hand point in the direction of the current; the thumb of that hand then point in the direction of the magnetic field.

Magnetic force can only change the direction of the particle’s motion, not its sound. 1 T = 1 kg/C-s The SI unit of magnetic field ( β ) is Tesla (T). A non-SI unit called gauss (G) is also used. 10 4 G = 1 T

Sample Problem: The velocity of an electron in a magnetic field of 2T is 4 x 10 5 m/s perpendicular to the field. Find the force that acts on the charge.

Magnetic Force Magnetic force on a current Since current, by definition are moving charge, current carrying conductors can also be moved by magnetic field. The magnetic force for a straight conductor in a uniform electric field. Where: F = magnetic force (Newton) I = current (ampere) L = Length of the conductor inside the magnetic field (meter) β = Magnetic field (Tesla)

The direction of the magnetic force is determined by right-hand-rule. The magnitude of magnetic force is: Where: θ = is the angle between the wire and the magnetic field.

Sample Problem: A wire 0.10 m long carrying a current of 2.0 A is at 30 0 angle with respect to the magnetic field. If the magnetic field strength is 0.20 T, what is the magnitude of the force on the wire?

Magnetic field of Earth Magnetic north pole is located somewhere in the Greenland, near but not exactly in the same location as geographic north pole. Magnetic south pole is at its direct opposite. Earth is the giant magnet that generates magnetic field. It enables compasses to work. Earth magnetic north pole is actually the “south pole”, where magnetic field terminates, and the magnetic south pole is actually the “north pole” from where the magnetic field emanates.

Compasses: Instrument used to find direction. Composed of slender bar magnet or low friction pivots. Follow the magnetic field lines of the earth. Point towards the geographic north pole.

Earth Magnetosphere Region that contains a mix of electrically charged particles. Electric and magnetic phenomena dominate rather than gravitational phenomena. Shield earth from the solar wind is called bow shock.

Van Allen Radiation Belts Traps high energy particles that leaked to magnetosphere. Regions of particularly high concentration of charged particles. Are responsible for the aurora (Northern and Southern Lights).

Conductors with Current Electric current generate magnetic field. Hans Christian Oersted noticed that the electric current can influence a compass needle. Oersed and Andre-Marie Ampere shows that current carrying wires exert force to one another.

Straight Conductor The direction of the magnetic field in a straight conductor can be determined using the right-hand-rule . β I Conductor

Single-Loop Conductor The conductor maybe in the shape of circle, ellipse or polygon. The magnetic field line’s direction must be according to the right-hand-rule with respect to the current.

The magnetic field vector at the center of the loop adds-up as one big magnetic field vector. I β

Solinoid Conducting wire coiled in the shape of helix. Function like several adjacent single-loop conductor. Similar to the wire coiled around an iron core (usually a nail) in an electromagnet.

The magnetic field vectors add-up at the center. C

An ideal solenoid is a solenoid of infinite length and uniform magnetic field inside the coil. A real solenoid is a solenoid of limit length. Its magnetic field is uniform near the center but not uniform near the ends.

Moving Charged Particles All magnetic fields are generated by charging electric fields. Moving charged particles generates electric field. Currents generate electric field because it is made-up of moving charge. + - Positive charge : Use Right-Hand-Rule Negative charge : Use Left-Hand-Rule

If a charge is moving relative to a point, the electric field at that point due to the charge is changing. This on-going change generates magnetic field. Note: Charging Electric field generates magnetic fields. E 1 E 2 E 3 + + + + + + E 1 E 2 E 3

Calculating the Magnetic Field Biot-Savart Law Where: µ o = 4µ x 10 -7 Tm/A = 1.26 x 10 -6 Tm/A r d β I dl

Different Conductor Configurations Long straight conductor: I β r

Long cylindrical conductor of radius R Outside the conductor* Inside the conductor* R r Outside Inside I β

Circular loop of Radius r Center of a circular arc with central angle Ø (in Radian) I r Ø

Distance z away directly above or below the center of circular loop. z r I

Long solenoid (almost Ideal) with number of turns ( N ) per unit length. Inside the solenoid and near the center Outside the solenoid N

Sample Problem: Straight conductor : What is the magnitude of the magnetic field 6.1 m below a power line in which there is a steady current of 100 A? Field along a solenoid: A solenoid of length 30.0 cm and radius 2.0 cm is closely winded with 200 turns of wire. The current in the winding is 5.0 A. Compute the magnetic field magnitude at a point near the center of the solenoid.

Parallel Current The force between two parallel current I a and I b is given by: Where: L = Length of the conductors d = distance between the conductors

The force is attractive if the currents are toward the same direction and repulsive if toward opposite directions. L I a I b d

Sample Problem: Parallel Currents Two long parallel wires are separated by distance of 8.0 cm. The current running along these wires are equal in magnitude but opposite direction. What is the current along the wires if the magnitude field halfway between them is 300.0 N? What is the force between the wires if the length of both of them is 4.0 m? Is this force attractive or repulsive?

Magnetic Materials Atoms are like tiny magnets. The electrons form a microscopic loop.

Atoms are like tiny magnets. The electrons form a microscopic loop. Moving electrons generate magnetic field. Hence, atoms are like small magnets. Most objects do not generate magnetic field despite being made-up of atoms because the atoms are oriented randomly: the atoms cancel each other’s magnetic field. + - I

Types of Magnetic Materials Paramagnetic Ferromagnetic Diamagnetic Assignment: Research “Types of magnetic materials” Computerized, Short Bond paper. To be submitted next meeting.

Field Symmetry Magnetic Flux defined The magnetic flux ( Φ β ) is the strength of an electric field over an area in a field region. Where: β = Magnetic field (Tesla, T) A = cross-sectional area (m 2 ) θ = Angle

The term Magnetic flux density is synonymous to magnetic field, defined as the magnetic flux per unit of perpendicular area The SI unit of magnetic flux is weber (Wb). 1 Wb = 1 T*m 2 N S

Conducting loop in a magnetic field Induction The process of producing current and emf by changing magnetic field. Induced current The current produced by changing magnetic field. Induced emf The work done per unit change in producing induced current.

Note: Changing magnetic field generates electric field. N S

Faraday’s Law States that the induced emf in a close loop equals the negative of the time rate of change of the magnetic flux through the loop.

Induced emf appears on the conducting loop if any of the following happens: The magnetic field is changing. The area of the loop within the magnetic field is changing. The conducting loop is rotating while immersed to magnetic field.

Sample Problem: A single loop of wire with an enclosed area of 6.00 cm 2 is in a region of uniform magnetic field, with the field perpendicular to the plane of the loop. The magnetic field is decreasing at a constant rate of 0.150 T/s. What is the induced emf ? If the loop has a resistance of 0.300 ohms what is the current induced in the loop?

Lenz’s Law States that the induced current runs to the direction in such a way that it generates magnetic field to oppose the changes in the magnetic flux that induced the current.

Used in determining the direction of induced current and induced emf . S N β S N β S N β β ind β ind I = 0 I I (A) (B) (C) No motion β increasing in the loop β decreasing in the loop

Problem Solving: A rectangular inductor of unknown length and width of 0.2 m moves at 12 m/s to the right. It is oriented perpendicular to a magnetic field of 0.4 T. What is the induced emf in the circuit? What is the direction of the induced emf? If the resistance across the loop is 0.3 ohms, What is the current?

Inductance Tendency of an electrical circuit to oppose the starting, stopping or changing the current. Its SI unit is henry (H): 1H = 1 Tm 2 /A

Inductor Provides inductance in a circuit. Produce uniform magnetic field. The inductance L of an inductor with number of turns N is given by:

Problem Solving: A current of 5.0 mA passess through the solenoid inductor with 400 turns and inductance of 8.0 mH. What is the magnetic flux through the coil?

Self-inductance happens when two adjacent turns of a solenoid inductor induced one another to changing electric current. The result of this is the intended function of the inductor: to resist changes in current. Self-inductance

The self-induced emf is the emf that arises due to the turns in the inductor inducing one another: Self induced emf opposes the current. I I CURRENT DECREASING CURRENT INCREASING ε L ε L ε L ε L

The self-induced emf can be solved using the formula:

The inductance does not oppose the current itself, only the change in current. It opposes both increase and decrease in current. Inductance L Inductor If the current is increasing then the voltage Opposing that Change is created By the magnetic Field of the coil.

Mutual-Inductance (M) Proportionality between the emf generated in a coil to the change in current in the other coil which produces it. Arises when to coils in close proximity induces emf to one another.

Notation: the subscript that the stand for the inducing coil comes second and the subscript that stands for the coil being induced comes first. Equation of induced emf: ε 21 is the emf induced in coil 2 due to change in current in coil 1 and ε 12 is the emf induced in coil 1 cue to change in current in coil 2.

Sample Problem: Two single-turn coils are fixed in location such that they can induced emf to one another. When the first coil has no current and the current in the second coil increases at rate of 15.0 A/s, the emf in the first coil is 25.0 mV. What is their mutual inductance? When the second coil has no current and the first coil has current of 3.60 A, What is the flux linkage in the second coil?

Alternating Current Alternating current (ac) The current is not constant, but varies sinusoidally with time. I t

Advantages over direct current (dc) Easier than transmit since charge carriers are not required to travel over long distance. Enables transformers to work by utilizing Faraday’s Law of induction. More readily adaptable to rotating machineries such as generators and electric motors.

Alternating Current Generator The emf ε varies sinusoidally with time: Driving frequency f d : Where: W d = Angle of frequency of the emf t = time ε m = amplitude of the emf

The current I varies sinusoidally with time:

Resistive Load The phase constant is zero. Time-varying voltage: Where: V R = voltage across the resistor V Rm = amplitude of the voltage Time-varying current: Where: I R = current through the resistor I Rm = amplitude of the current Relation of amplitude of current and voltage: R I I ε

Capacitive Load The phase constant is Time-varying voltage Where: V c = voltage across the Capacitor V c m = amplitude of the voltage Time-varying current: Where: I c = current through the Capacitor I c m = amplitude of the current C I I ε

Relation of amplitude of current and voltage: The Quantity X C is called capacitive reactance: The unit of reactance is ohms ( Ω )

Inductive Load The phase constant is: Time-varying voltage Where: V L = voltage across the inductor V L m = amplitude of the voltage Time-varying current: Where: I L = current through the inductor I L m = amplitude of the current I I ε L

Relation of amplitude of current and voltage: The Quantity X L is called inductance reactance: The unit of reactance is ohms ( Ω )

RLC Series Circuit The Resistor, Inductor and Capacitor are connected in series with ac emf device. ε L R C I I I I

Voltage and current in series circuit: Relation of the amplitudes of voltage and current:

The quantity is called Impedance (Z): The unit of reactance is ohms ( Ω ). The phase constant can be solved using the equation:

Sample Problem: A 160 Ω resistor, 15.0 µF capacitor and 230 mH inductor are connected to form RLC circuit with an ac generator whose conducting loop rotates at 60.0 full rotation per second and with emf amplitude of 36.0 V. Find: The impedance of the circuit. The current flowing through the circuit. The phase constant

Transformer A device used to change the voltage and current levels in an AC circuit. Step-up transformer: V out > V in Step-down transformer: V in > V out

input output Primary winding Secondary winding core Magnetic flux Transformer

Characteristic of an Ideal Transformer Has two coil or windings, electrically insulated from each other but wound on the same core. Core typically made-up of material with large relative permeability such as iron. Primary coil (input) – winding to which power supply is received. Secondary coil (output) – winding to which power is delivered. Resistance is negligible. Magnetic field is confined to the iron core.

Transformation of voltage and current The induced emf in primary, ε 1 , and secondary coil, ε 2 are: We can combine the equation above as:

Terminal voltage of primary and secondary coil: Step-up transformer, ε 2 > ε 1 and N 2 > N 1 Step-down transformer, ε 1 > ε 2 and N 1 > N 2

Power of Transformer: If we place a resistance, R, to complete the circuit in the secondary coil:

Problem Solving: A transformer has 100 turns on its primary coil and 300 turns on the secondary coil. If the primary voltage is 110.0 V and primary current is 5.00 A. What are the secondary voltage and current?

Waves A disturbance that travels through a material medium. Carries energy. Can transfer energy from one place to another without actual motion of an object or particle. Some waves can travel through vacuum and do not require a material medium; Example is Light

Types of Waves Transverse wave The motion of the particles at the moment the disturbance passes through is perpendicular to the propagation of the wave. Example: Light

Wave propagation Wave propagation Wave propagation Particle motion Particle motion Particle motion Undisturbed position Undisturbed position Undisturbed position

Longitudinal Wave The motion of the particles at the moment the disturbance passes through is linear to the propagation of the wave. Example: Sound

Properties of Wave Wavelength The distance between two adjacent particles or points that behave in the same manner. The unit is meter (m). Denoted by Greek letter (“lamda” λ )

Period The time it takes for one complete wavelength to pass through a certain point. The unit is second (s). Denoted as capital T.

Frequency The number of wave passing through a certain point per unit time. The unit is per second or hertz (/s or s -1 ) Reciprocal of Period (T) Denoted by small letter f .

Amplitude The maximum displacement of particle due to disturbance before returning to its undisturbed position. The unit is meter (m). Denoted by capital letter A.

Speed Distance traveled by the disturbance per unit of time. The Unit is meter per second (m/s). Denoted by small letter v.

m/s v speed Meter (m) A amplitude /s, s -1 Hertz (Hz) f frequency Second (s) T Period Meter (m) λ Wavelength relation unit Symbol Quantity

Problem Solving: The frequency of a wave traveling across the string is 0.167 Hz and its wavelength is 9.00 cm. What is the period and speed of the wave?

Behavior of Wave Refraction It is the change in the wave’s direction as it crosses the boundaries between two medium Incident Refracted Medium 1 Medium 2

Reflection It is the change in the wave’s direction without crossing to the adjacent medium. Incident Reflected Medium 1 Medium 2

Interference It is the combination of two or more waves as they pass through the same location at the same time.

Two kinds of Interference Constructive interference It happens when the waves passing through the same location are in-phase, resulting to a combined disturbance with higher amplitude.

Destructive interference It happens when the waves passing through the same location are out-phase, resulting to a cancelled disturbance with lower or zero amplitude.

Diffraction It is the spreading of wave after it passed through a small slit. Each point at the wavefront acts as tiny source of smaller waves called wavelets. The interference among wavelets keep wave in shape. However, If the slit is small enough, some of the wavelets will not be able to pass through; with no interference, the wavelets from one point will be able to propagate. wavefronts slit obstacle Diffracted wave

Electromagnetic Wave Changing electric field creates magnetic field. Changing magnetic filed creates electric filed.

Electromagnetic waves are disturbance produced by propagating electric and magnetic fields. Speed in a vacuum : (299,792,458 m/s or 3.00 x 10 8 m/s) Examples: Light Infrared rays Ultraviolet rays Radio waves rays

Light is the electromagnetic wave that is visible to the eyes, with wavelengths between 4 x10 -7 m and 7 x10 -7 m and frequencies between 7 x 10 14 hertz and 4 x10 14 hertz. Electric field Magnetic field Direction Wavelength

Speed in a vacuum Where: c= speed of the electromagnetic waves (m/s) E=electric field (V/m) β = magnetic field (Weber/m 2 ) ε o = permitivity constant μ o = permeability constant

Problem Solving: At a particular time the magnetic field intensity in electromagnetic wave is 2 x10 -10 Wb/m 2 . Calculate the magnitude of the electric field intensity.

Electromagnetic Spectrum Assignment: Computerized, Short bond paper Research work Gamma Rays X-rays Ultraviolet rays Visible Light Infrared Radio wave

Wavelength and frequency The frequency and wavelength of electromagnetic (EM) wave are inversely proportional. Where: f = frequency of the wave (Hz) c = speed of the EM wave in a vacuum λ = wavelength (m)

Problem Solving: What is the range of the wavelength of the visible light? (see… Electromagnetic spectrum table)

Visible Light Form of electromagnetic radiation that our eyes detect. Wavelength ranging from 400 to 760 nm. People are able to “see” an object ligth enters the eyes. Represented as: Ray a thin beam of light that travel in a straight line. Wavefront It is the line (not necessarily straight) or surface connecting all the light that left a source at the same time. Can be reflected and refracted.

Optics Branch of electromagnetism that deals with the nature, properties and behavior of light. Two branches: Geometric Optics Describes light propagation in terms of rays. Physical Optics Treat light propagation as a wave phenomenon rather than a ray phenomenon.

Reflection of Light The angle of incident ( θ i ) is equal to the angle of reflection ( θ r )

θ i θ r θ i = 0 θ r = 0 Mirror A The light is parallel to The plane of mirror. No Reflection. Mirror B Light is reflected at an angle. θ i = θ r A B Mirror C Incident and reflected Light are both perpendicular To the plane of mirror. θ i - θ r =0 C

Refraction of Light The refraction of light is governed by Snell’s Law: Where: θ i = angle of incident ray θ r = angle of refraction ray n i & n r = indices of refraction

AIR WATER Incident ray Refracted ray θ i θ r

Index of refraction (n) The ratio of light’s speed in a vacuum and light’s speed in that material. Property of the material has no unit. Where: c = 3.00 x10 8 m/s light in vacuum. v = speed of light in the medium.

Sample Problem: A light beam crosses from the vacuum to water with incident beam angle of 25.0 0 . If the index of refraction of vacuum is 1 and that of water is 1.33, what is the angle of refracted light beam?

Mirrors Surface that reflects so much light that they formed images. Made of polish metal (silver or copper) or glass with silver colored coating.

Two kinds of mirror: Plane mirror Flat surface and the reflected parallel light rays remain parallel. Plane mirror

Spherical mirror Its surface form a part of the surface of the sphere: Concave mirror – focuses reflected rays. Convex mirror – scatters reflected rays. Concave mirror Convex mirror Principal Axis Principal Axis

Parts of a mirror Principal axis An imaginary line passing through the center of the sphere and passing through the exact center of the mirror. Center of Curvature (C) The point in the center of the sphere from which the mirror was sliced.

Vertex (V) The point on the mirror’s surface where the principal axis meets the mirror. The geometric center of the mirror. Focal point (F) A point between the vertex and the center of the curvature. In concave mirrors, this is the point where the reflected rays intersect. In convex mirror, this is the point from where the reflected ray apparently originates.

Radius of the curvature (R) Distance from the vertex to the center of the curvature. The radius of the sphere from which the mirror was cut. Focal Length (f) The distance from the mirror to the focal point. One-half the radius of the curvature. For concave mirrors: For convex mirrors:

Problem Solving: Light from the distance is collected by a concave mirror. How far from the mirror do the light rays converge if the radius of curvature of the mirror is 200 cm.?

Mirrors Image Formation In Plane mirror In order to see the image of an object in a mirror: You must view at the image; When you view at the image, light will come to your eyes along that line of sight. The image location is located at that position where observers are viewing the image of an object. It is the location behind the mirror where all the light appears to diverge from.

An image is formed because light emanates from the object in a variety of directions. Some of this light reaches the mirror and reflects off the mirror accordingly to the law of reflection.

In Concave mirror There are two types of image: Real image Light passes through the image’s location. Its formed when p > f Virtual image Light does not pass through the image’s location. Its formed when p < f

In Convex mirror The image is always virtual.

V Mirror C F For Concave Mirror Principal axis R f Object

V Mirror C F For Convex Mirror Principal axis f Object

Mirror Equation Where: f = focal length p = Object distance Distance from the object to the mirror. q = Image distance Distance from the image to the mirror. m = Magnification of the mirror y = size of the object y’ = size of the image

V Mirror C For Concave Mirror Principal axis p f Object q F Image

Ray Diagram Method (RDM) For Concave mirror: First Ray – Parallel to the axis, reflects to the mirror, then passing through focal point. Second Ray – Passing through the focal point, reflect to the mirror, then parallel to the axis. Third Ray – Passing through center of curvature, then bounce back.

RDM V Mirror C Principal axis Y F Image 1 st Ray 2 nd Ray 3 rd Ray Object Concave Mirror

Problem Solving: A 3.0 cm tall light bulb is placed a distance of 40 cm from a concave mirror having a focal length of 10.2 cm. Determine the image distance.

LENSES It is an optical system with two refracting surfaces.

Thin Lens Has two spherical surfaces close enough so its thickness can be neglected. Types of Thin lens Converging Lens Diverging Lens

Converging Lens Different kinds of converging lens: Meniscus Piano-convex Double-convex

Diverging Lens Different kinds of diverging lens: Meniscus Piano-concave Double-concave

Parts of Lenses Optic axis The central horizontal line defined by the centers of curvature of the two spherical surfaces. Focal points (F 1 and F 2 ) Focal Length (f) Distance between a focal point and the center of the lens The two focal lengths are always equal for a thin lens

Image Formation by thin lenses Real image are located on the side of the lens opposite that of the object. Virtual images are located on the same side of the lens as the object

Converging lens F 1 F 2 f f Optic axis

When a beam of parallel rays pass through the lens, the rays converge at one focal point. Its focal length is defined to be positive. F 2 F 1 f f F 2 F 1 f f Optic axis Optic axis

Converging lens can form either real or virtual image. Real image if the distance of the object from the center of the lens is greater than the focal length (p>f). Virtual image if the distance of the object from the center of the lens is less than the focal length (p<f). Image formation by Thin Lenses

Real Image F 1 F 2 Object Image f f p q Optic axis

Virtual Image F 1 Object F 2 f f Image p q Optic axis

Diverging lens F 1 F 2 f f Optic axis

When a beam of parallel rays are incident on this lens, the rays diverge after refraction. Its focal length is defined to be negative. F 2 F 1 f f F 2 F 1 f f Optic axis Optic axis

Image formation by Thin Lenses F 1 F 2 Object Image f f p q The image formed by a diverging lens is always virtual. Optic axis

Lens Equation Thin Lens equation: Lateral magnification:

Problem Solving: A converging lens has a focal length of 12.0 cm for an object 20.0 cm to the left of the lens, Determined: The image position The image magnification

Lenses and Mirror Assignment: Computerized, Short bond paper Research work The Eye The Camera The Magnifying Glass The Telescope (Galileo) The Microscope

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