CHECKPOINT: What is the current direction in this loop? And which side of the loop is the north pole?

A. Current clockwise; north pole on top

B. Current clockwise; north pole on bottom

C. Current anticlockwise; north pole on top

D. Current anticlockwise; north pole on bottom

Answer: B.

Magnetic force between parallel wires (p445)

The magnetic field B1 due to current I1 is perpendicular to current I2.

The force on current I2 is towards current I1.

There is an equal an opposite force exerted by current I2 on I1. The wires thus attract each other.

122 2dF I dl B

If current I1is reversed, B1 would be in opposite direction. Therefore we find that antiparallel currents repel.

Last lecture

Gauss’ law for magnetism

Got it? Which could be a magnetic field?

We found for highly symmetric charge distributions, we could calculate the electric field more easily using Gauss’s Law than Coulomb’s Law.

Similarly Ampère’s Law relates

the tangential component Bt of the magnetic field summed (integrated) around a closed curve C to

the current IC that passes through any surface bounded by C.

26.8 Ampère’s law

Iencircled is the net current that penetrates the surface bounded by the curve C.

0 enclosedcosC

B ds B ds I As in Gauss’s law, the field that appears in the integral is the net field arising from all sources, not just the encircled currents.

We can now use Ampère’s Law for situations where symmetry allows us to simplify, and solve the integral for the magnetic field.

C

B dl B dl

The simplest case is for an infinite long, straight, current-carrying wire.

Choose the circular curve centred on wire. Magnetic field is tangent to this circle from Biot-Savart law, and has the same magnitude all around the loop.

0 enclosed

0 enclosed

2

2

C

B d l B dl B R i

iB

R

If the length of the wire approaches infinity in both directions, we found

0

2 2

2

4

I aB

x x a

Compare with earlier result using Biot-Savart:

Magnetic field B due to a current in a straight wire.

1. The magnetic field due to a current element is parallel to the current element.

2. The magnetic field due to a current element varies inversely with the square of the distance from the element.

3. The magnetic field due to a long wire varies inversely with the square of the distance from the wire.

4. Ampere’s law is valid only of there is a high degree of symmetry.

A.True F. False

True

False

False

False

Magnetic fields

Solar currents

Example 26.7 Ampere’s law: solar currents

Applications of Ampere’s Law

Ex 26.8 Field outside and inside a long straight current-carrying wire

Ex 26.9 Field outside a current sheet

Ex 26.10 Field of a solenoid

Fig 26.39 Field of a toroidal solenoid

EXAMPLE 26.8: A long straight wire of radius R carries a current I that is uniformly distributed over the circular cross section of the wire. Find the magnetic field both (a) outside and (b) inside the wire.

I

I