Ampere's law relates the magnetic field around a closed loop to the electric current passing through the loop. It states that the line integral of the magnetic field around the loop equals the net current passing through the loop times the permeability of free space. The document provides examples of using Ampere's law to calculate magnetic fields for various current carrying wire configurations, including inside and outside a long straight wire, inside a solenoid, and inside and outside a toroidal coil.

1. Ampere’s Law

2. Ampere’s LawGauss’ law allowed us to find the net electric field due to any charge distribution (with little effort) by applying symmetry.Similarly the net magnetic field can be found with little effort if there is symmetry using Ampere’s law.

3. Ampere’s LawAmpere’s law,Where the integral is a line integral.B.ds is integrated around a closed loop called an Amperian loop.The current ienc is net current enclosed by the loop.

4. Ampere’s Lawie,ie ienc

5. Ampere’s LawThe figure shows the cross section of 3 arbitrary long straight wires with current as shown.

6. Ampere’s LawTwo of the currents are enclosed by an Amperian loop.

7. Ampere’s LawDirection ofintegrationAn arbitrary direction for the integration is chosen.

8. Ampere’s LawDirection ofintegrationThe loop is broken into elements of length ds (choose in the direction of the integration).Direction of B doesn’t need to be known before the integration!

9. Ampere’s LawDirection ofintegrationB can be in an arbitrary direction at some angle to ds as shown (from the right hand grip rule we know B must in the plane of page).We choose B to be in the direction as ds.

10. Ampere’s LawThe right hand screw (grip) rule is used to assign a direction to the enclosed currents.A current passing through the loop in the same direction as the thumb are positive ( in the opposite direction -ve).

11. Ampere’s LawConsider the integral,Direction ofintegration

12. Ampere’s LawApplying the screw rule,Direction ofintegration

13. Ampere’s LawExample. Find the magnetic field outside a long straight wire with current.Ir

14. Ampere’s LawWe draw an Amperian loop and the direction of integration.Amperian LoopWire surfaceDirection ofIntegration

15. Ampere’s LawRecall,Therefore,The equation derived earlier.

16. Ampere’s LawThe positive sign for the current collaborates that the direction of B was correct.

17. Ampere’s LawExample. Magnetic Field inside a Long Straight wire with current.Amperian LoopWire surface

18. Ampere’s LawAmpere’s Law,

19. Ampere’s LawAmpere’s Law,The charge enclosed is proportional to the area encircled by the loop,

20. Ampere’s LawThe current enclosed is positive from the right hand rule.

21. Applications of Ampere’s Law

22. Applications of Ampere’s lawLong-straight wireInsider a long straight wireToroidal coilSolenoid

23. Toroidal Coil

24. Toroidal CoilrI0Toroid has N loops of wire, carrying a current I0Ampere Loop, circle radius r No current flowing through loop thus B = 0 inside the Toroid

25. Toroidal CoilrAmpere Loop, circle radius r I0For each wire going in there is another wire comeing out Thus no nett current flowing through loop thus B = 0 outside the Toroid

26. Toroidal CoilrI0ZoomToroid has N loops of wire For each loop of the coil an extra I0 of current passes through the Ampere LoopAmpere Loop, circle radius r

27. Solenoid

28. Infinitely Long SolenoidWire carrying a current of I0wrapped around with n coils per unit lengthZoom looks very similar to the toroid with a very large radius

29. rToroidal Coil: RevisitedI0Toroid has N loops of wire, carrying a current I0Central radius R circumference is 2pRNumber of coils per unit length n isFrom earlier:Independent of R

30. Infinitely Long SolenoidWire carrying a current of I0wrapped around with n coils per unit lengthField at centre is same as torus of infinite radius