Week 6

Question 1

Magnetic part of Lorentz force law

Answer 1

Magnetism analogue of Coloumb’s law

Which is force on test charge Q, moving with velocity v in magnetic field B

Question 2

Why are the velocities in magnetostatics?

Answer 2

The movement of charges causes a static magnetic field (unchanging magnetic field)

Question 3

Right hand rule

Answer 3

For cross product eg V x B

First point thumb in direction of V, then index finger in direction of B

Resultantly middle finger points in direction of cross prod vector

Each subsequent finger is orthogonal

Question 4

Lorentz force law

Answer 4

Question 5

Prove that the force on a closed loop of wire carrying current I due to a constant magnetic field B is zero

Answer 5

Question 6

The size of force due to a magnetic field is?

Answer 6

Where α is the angle in between v and B

The field B produces no force on the static test charge Q
Therefore the size of force depends only on |v|Sinα

Question 7

Interesting consequence of force law

Answer 7

Direction of force depends on sign of charge

You can tell the charge of a particle by seeing which direction it is deflected in when moving through a magnetic field

Question 8

Magnetic forces and work

Answer 8

F_magnetic is perpendicular to v therefore force due to magnetic field does not do any work as

F_magnetic • dl = F_magnetic • vdt = 0

Magnetic forces can’t accelerate a charged particle, only alter direction that test charges move

Question 9

A current is

Answer 9

A charge in motion

Question 10

Units for current

Answer 10

1 Ampere = 1 coulomb per second

Question 11

Current vector points

Answer 11

In the direction opposite to the movement of electrons

Question 12

Calculate F_magnetic for charge density λ and on a length of wire

Answer 12

Question 13

Define the magnetic force on a surface

Answer 13

Question 14

Define a magnetic force through a volume

Answer 14

Question 15

Derive the continuity equation

Answer 15

Taking the volume current density (which ishow much charge flows through a surface area element):
J • da = ρvCos(θ)da

Which then in time is ρ(vCos(θ)dt)da then

Question 16

Steady current

Answer 16

No build up of charge

Question 17

Assumptions of electrostatics and magnetostatics

Answer 17

accordingly

stationary charge
steady current

Question 18

Significance of?

Answer 18

This is a consequence of working with magnetostatics

This means the field lines of J can’t begin or end on any local point (compare to grad • E = ρ/ε0 in electro case where electric field lines can begin or end on any charges which give rise to nonzero value of ρ)

Question 19

Biot Savart rule

Answer 19

Physical rule that governs how steady currents in wires produce static magnetic fields

Where:
I = Idl’ for a current wire
P is path along wire
s = r - r’

Question 20

μ₀ ?

Answer 20

4π * 10**-7

Question 21

Right hand rule for magnetic field

Answer 21

If you wrap your hand around a wire such that your thumb is pointing in the direction of flow of the current through that wire (up or down) then

The direction that your fingers curl around the wire represent that direction of flow of the magnetic field at some arbitrarily close distance

Question 22

Trig substitutions

Answer 22

Question 23

Sin rule

Answer 23

Question 24

Cosine rule

Answer 24

Question 25

Angular basis vector in cylindrical coordinates

Answer 25

Question 26

Find magnetic field at a fixed distance ρ from an infinite straight wire on the z axis carrying steady current I travelling up in z direction

Answer 26

Where **r** is a point at radial distance ρ from the wire

Question 27

When does a charge NOT contribute to the flux integral of an electric field

Answer 27

When the charge is located outside of the surface being integrated over

Question 28

When does a current not contribute to the line integral of a closed path

Answer 28

When the path doesn’t enclose the wire

Question 29

Show the curl of **B** around the infinitely long straight wire coinciding with z axis

Answer 29

Question 30

Deduce that

Answer 30

Given that rhs of stokes theorem = μ₀*I _P

Question 31

State Biot Savart law in terms of volume current density **J**

Answer 31

Question 32

Use Biot Savart to deduce

Answer 32

Question 33

Derive Ampere’s law

Answer 33

Question 34

Magnetic field as of an infinite charged plane

Answer 34

Where K is the uniform surface current

Question 35

Motivate the vector potential

Answer 35

Given M3, it would be sufficient to show that **B** can be given as a curl of a vector field This is because the divergence of the curl = 0

Question 36

Find Kronecker deltas from epsilon tensors

Answer 36

Question 37

Laplacian in components

Answer 37

Question 38

Is magnetic potential uniquely defined

Answer 38

No, it is defined up to a gradient (as opposed to constant for electro)

Question 39

Derive the magneto equivalent of poisons equation with solution

Answer 39

Question 40

Deduce vector potential for **B** produced by an infinitely long wire carrying charge **I** = I**e_z** and coinciding with z

Answer 40

Question 41

Derive the first 2 terms of multiple expansion for magnetic field

Answer 41