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.

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.

encisdB 0.

Ampere’s Law

• ie,

• ie ienc

N

n

nisdB1

0.

N

nni

1

Ampere’s Law

• The figure shows the cross section of 3 arbitrary long straight wires with current as shown.

1i

3i 2i

Ampere’s Law

• Two of the currents are enclosed by an Amperian loop.

1i

3i 2i

Ampere’s Law

• An arbitrary direction for the integration is chosen.

3i

1i

2i

Direction ofintegration

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!

3i

1i

2i

Direction ofintegration

sd

B

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.

3i

1i

2i

Direction ofintegration

sd

B

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,

3i

1i

2i

Direction ofintegration

sd

B

N

n

nisdB1

0. dsB cos

Ampere’s Law

• Applying the screw rule,

3i

1i

2i

Direction ofintegration

sd

B

210cos iidsB

Ampere’s Law

• Example. Find the magnetic field outside a long straight wire with current.

r

I

Ampere’s Law

• We draw an Amperian loop and the direction of integration.

Wire surface

Amperian Loop

Direction ofIntegration

B

sd

0

Ampere’s Law

• Recall,

• Therefore,

• The equation derived earlier.

N

n

nisdB1

0.

rBdsBdsB 2cos IrB 02

r

IB

2

0

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 Loopr

R

B

sd

Ampere’s Law

• Ampere’s Law,

N

n

nisdB1

0.

rBdsBdsB 2cos

Ampere’s Law

• Ampere’s Law,

• The charge enclosed is proportional to the area encircled by the loop,

N

n

nisdB1

0.

rBdsBdsB 2cos

iR

rienc 2

2

Ampere’s Law

• The current enclosed is positive from the right hand rule.

2

2

02R

rirB

rR

iB

20

2