Week 7

Question 1

Derive EMF (electromotive force) in rectangular loop of wire moving into magnetic field

Answer 1

Reduce Lorentz to find F_magnetic = Q(v x B)

In components this becomes
F_magnetic = Q((-v e_x) x (-B e_z)) = - QvBe_y on each charge Q

The Lorentz force only does work on the side of the wire that is entirely within the field (and perpendicular to the motion of the rectangular loop)

Then the work down around the loop of wire is:

Question 2

Use the EMF to deduce Faraday’s law

Answer 2

Integrating both sides wrt time across an element of wire dl across a distance vdt we find (dl x v)dt = da

Hence

Question 3

Summarise Faraday’s equation

Answer 3

An electromotive force is produced around a loop P whenever the magnetic flux through the surface S, bounded by P, changes.

Question 4

Lenz’s law

Answer 4

The induced current will flow in the direction such that the magnetic flux it induces will aim to reduce the initial magnetic flux

Question 5

Modify Ampere’s law (by Maxwell)

Answer 5

Question 6

The displacement current

Answer 6

The term added by Maxwell to ampere’s law

Question 7

What is a capacitor

Answer 7

Pair of parallel equal and oppositely charged conducting plates

Question 8

Explain how

Answer 8

Question 9

Maxwell’s equations in a vacuum

Answer 9

When ρ = 0 and J = 0

Question 10

Decouple E and B

Answer 10

In an electromagnetic vacuum

WE DO NOT NEED TO DERIVE THIS, JUST USE:

Question 11

Monochromatic plan wave solutions to Maxwell’s equations

Answer 11

WE DO NOT NEED TO DERIVE THIS, JUST USE:

Uniform over a plane, focus on a single ω

k^ is the normalised direction of travel of the wave

k • n^ = 0