Ampere’s Law

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.

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.

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.

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.

Ampere’s Law ie, ie ienc

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

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

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

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

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

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).

Ampere’s Law Consider the integral, Direction of integration

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

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

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

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

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

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

Ampere’s Law Ampere’s Law,

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

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