Ampere’s Law.pptx
- 1. Ampere’s Law: It is stated as “ The line integral of magnetic field 𝐵 along a closed path is equal to 𝜇𝑜 times the total current ‘i’ enclosed by the closed path” and expressed as 𝐵. 𝑑𝑠 = 𝜇𝑜𝑖 This is known as Ampere’s Law. In Ampere’ Law, we draw an Amperian Loop that is a imaginary closed curve to find magnetic field B.
- 2. Application of Ampere’s Law: A Long Straight Wire: Consider a long, straight wire of length L and small thickness. We have to apply Ampere’s Law to find the magnetic field at a distance d. We choose an Amperian loop as a circle of radius ‘d’ centered on the wire with its plane perpendicular to the wire. Assume that B is constant around the path.
- 3. We know that from oersted’s experiment that 𝐵 has only a tangential component. Thus the angle 𝜃 is zero and the line integral becomes; 𝐵. 𝑑𝑠 = 𝐵𝑑𝑠 cos 𝜃 = 𝐵𝑑𝑠 cos 0 = 𝐵 𝑑𝑠 Where 𝑑𝑠 = 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒 = 2𝜋𝑑
- 4. So, the Ampere’s Law gives 𝐵𝑑𝑠 cos 𝜃 = 𝜇𝑜𝑖 𝐵 𝑑𝑠 = 𝜇𝑜𝑖 𝐵 2𝜋𝑑 = 𝜇𝑜𝑖 𝐵 = 𝜇𝑜𝑖 2𝜋𝑑
- 5. A solenoid: Consider an ideal solenoid and choose an Amperian loop in the shape of rectangle abcda. Assume that magnetic field is parallel to axis of this ideal solenoid and constant in magnitude along line ab. Field is also uniform in the interior.
- 6. Consider Ampere’s Law; 𝐵. 𝑑𝑠 = 𝜇𝑜𝑖 The left side of Ampere’s Law can be written as the sum of four integrals, one for each path segment: 𝐵. 𝑑𝑠 = 𝑎 𝑏 𝐵. 𝑑𝑠 + 𝑏 𝑐 𝐵. 𝑑𝑠 + 𝑐 𝑑 𝐵. 𝑑𝑠 + 𝑑 𝑎 𝐵. 𝑑𝑠 Where 𝑎 𝑏 𝐵. 𝑑𝑠 = 𝑎 𝑏 𝐵𝑑𝑠 cos 0 = 𝐵 𝑎 𝑏 𝑑𝑠 = 𝐵ℎ
- 7. 𝑏 𝑐 𝐵. 𝑑𝑠 = 𝑑 𝑎 𝐵. 𝑑𝑠 = 0 Where 𝑏 𝑐 𝐵𝑑𝑠 cos 90 = 0 𝑐 𝑑 𝐵. 𝑑𝑠 = 0 Because 𝐵 is zero for all external points for an ideal solenoid. So, for entire rectangular path, 𝐵. 𝑑𝑠 = 𝐵ℎ + 0 + 0 + 0 = 𝐵ℎ
- 8. The net current I that passes through the rectangular amperian loop is not the same as current in the solenoid because winding pass through the loop more than once. Let ‘n’ be the no. of turns per unit length, then ‘nh’ be the no of turns inside the loop and the total current through the Amperian loop is 𝑖′ = 𝑛ℎ𝑖 So, Amperian’s Law becomes 𝐵. 𝑑𝑠 = µ𝑜𝑖′ 𝐵ℎ = µ𝑜𝑛ℎ𝑖 𝐵 = µ𝑜𝑛𝑖 That is the final expression.
- 9. A Toroid: “ A toroid may be consider to be a solenoid bent into the shape of a doughnut”. We can use Ampere’s Law to find the magnetic field at interior points. From symmetry, the lines of 𝐵 from concentric circles inside the toroid. Let us choose a concentric circle of radius ‘r’ as an Amperian loop and transverse it in the clockwise direction. Ampere’s Law yields 𝐵 2𝜋𝑟 = 𝜇𝑜𝑖𝑁
- 10. Where ‘i’ is the current in the toroid windings and N is the total number of turns. This gives 𝐵 = 𝜇𝑜𝑖𝑁 2𝜋𝑟 To contrast to the solenoid, B is not constant over the cross-section of the toroid. From Ampere’s Law, B=0 for points outside an ideal toroid and in the central cavity ‘2𝜋𝑟’ is the central circumference of the toroid and 𝑁 2𝜋𝑟 Is just n , i.e; no of turns per unit length, so by substituting, we get 𝐵 = µ𝑜𝑛𝑖 The direction of M.F within a toroid follows from the right hand rule; curl the finger in your right hand in the direction of current and your thumb points the direction of Magnetic field.
- 11. Chapter 8 Faraday’ Law of Induction
- 12. Magnetic Flux: Φ𝐵 “ It is the change in the number of field lines passing through a circuit loop that induces the emf in the loop”. Specifically, it is the rate of change in the number of field lines passing through the loop that determines the induced emf. To make this statement quantitative, we introduce the magnetic flux Φ𝐵 that is “the magnetic flux is the measure of the number of field lines passing through the surface.” Φ𝐵 = 𝐵. 𝑑𝐴 Where 𝑑𝐴 is area element of surface.
- 13. Where Φ𝐵 = 𝐵. 𝐴 Φ𝐵 = 𝐵𝐴 cos 𝜃 Where 𝜃 is the angle between the normal to the surface and their direction of the field. The SI unit of magnetic flux is the weber where 1 weber = 𝑡𝑒𝑠𝑙𝑎. 𝑚𝑒𝑡𝑒𝑟2
- 14. Faraday’s Law of Induction: “ The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is changing with time”. Mathematically; ξ = − 𝑑Φ𝐵 𝑑𝑡 Where ξ is induced emf. Where 1 𝑣𝑜𝑙𝑡 = 1 𝑤𝑒𝑏𝑒𝑟/sec If the coil consists of N turns , then ξ = 𝑁 𝑑Φ𝐵 𝑑𝑡
- 15. Lenz’s Law: • We find the direction of the induced emf based on the (induced) current that it would produced, using a rule proposed in 1834 by Heinrich Lenz and known as Lenz’s law: “The flux of the magnetic field due to the induced current opposes the change in flux that causes the induced current”. Lenz’s law refers to induced current which means that it applies only to closed conducting circuits. If the ‘change in flux’ is an increase then lenz’s law requires that the direction of the induced current oppose the increase and vice versa.
- 16. Gauss’s Law For Magnetism: It stated as “ The net Flux of the magnetic field through any closed surface is zero”. Mathematically, Φ𝐵 = 𝐵. 𝑑𝐴 = 0 i.e; no magnetic monopoles exists.