1. Ampere’s Law

2. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law.
Define Ampere’s Law.
Show how to use Ampere’s Law for cases with symmetry.

3. Student Objectives
Recognise Ampere’s Law to be analogous to Gauss’ Law.
Recognise similar concepts: (1) draw an imaginary shape enclosing the current carrying conductor, (2) current enclosed.

4. Ampere’s Law
Gauss’ 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.

5. Ampere’s Law Ampere’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.

6. Ampere’s Law
ie,
ie ienc

7. Ampere’s Law
The figure shows the cross section of 3 arbitrary long straight wires with current as shown.

8. Ampere’s Law
Two of the currents are enclosed by an Amperian loop.

9. Ampere’s Law An arbitrary direction for the integration is chosen.
Direction of
integration

10. Ampere’s Law
The 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!
Direction of
integration

11. Ampere’s Law
B 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.
Direction of
integration

12. Ampere’s Law
The 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).

13. Ampere’s Law
Consider the integral,
Direction of
integration

14. Ampere’s Law
Applying the screw rule,
Direction of
integration

15. Ampere’s Law
Example. Find the magnetic field outside a long straight wire with current. I r

16. Ampere’s Law
We draw an Amperian loop and the direction of integration.
Amperian Loop
Wire surface
Direction of
Integration

17. Ampere’s Law
Recall,
Therefore,
The equation derived earlier.

18. Ampere’s Law
The positive sign for the current collaborates that the direction of B was correct.

19. Ampere’s Law
Example. Magnetic Field inside a Long Straight wire with current.
Wire surface
Amperian Loop

20. Ampere’s Law
Ampere’s Law,

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

22. Ampere’s Law
The current enclosed is positive from the right hand rule.