Sections 19-22

Question 1

What 3 things can we assume about a solenoid?

Answer 1

• Long so we don’t need to worry about end conditions
• Number of turns per unit length n >> 1 so can assume axial symmetry
• Wire wound forward and back so no net current along axis

Question 2

What are the 4 steps to calculating the magnetic field around a solenoid?

Answer 2

1. Apply Amperes law to the path - find there is no azimuthal field
2. Apply solenoidal condition - no radial field
3. Choose rectangular path outside the solenoid and use Amperes law for each side of the path - find the axial field outside the solenoid is zero
4. Choose rectangular path dipping into the solenoid, current inside is NI, then use Amperes law to find the field inside solenoid is: B = μ0*n*I k

Question 3

What is the magnetic dipole moment μ defined as?

Answer 3

μ = IA n(hat)

Question 4

What does the current in μ generate itself in a dipole?

Answer 4

It generates a magnetic field which is parallel to μ (the axis of the current loop)

Question 5

What does the torque in a dipole do?

Answer 5

It acts to align μ and B0, where μ is the normal vector to the current loop multiplied by IA.

Question 6

What can atoms be said to behave like? What does this create?

Answer 6

Magnetic dipoles. If there are many small dipoles (atoms), it creates a surface current, and the current inside the material is zero.

Question 7

What happens if you place a material into an external magnetic field B0?

Answer 7

B0 induces a μ and forces the atoms in the material to align to B0, and the atoms then create their own magnetic field.

Question 8

When a material is exposed to an external magnetic field B0, what happens to the magnetic field inside the material?

Answer 8

It becomes B = μr B0, where μr is the relative permeability which is a property of the material.

Question 9

What are the 4 types of magnetic materials?

Answer 9

• Diamagnetic
• Paramagnetic
• Ferromagnetic
• Hard Ferromagnetic

Question 10

What is a diamagnetic material?

Answer 10

μr < 1, so the B-field inside the material is reduced.

Question 11

What is a paramagnetic material?

Answer 11

μr > 1, so the B-field inside the material is increased.

Question 12

What is a ferromagnetic material?

Answer 12

μr >> 1, so the B-field inside the material is greatly increased.

Question 13

What is a hard ferromagnetic material?

Answer 13

Permanent magnets which maintain a magnetic field even when there is no external magnetic field. μr not defined.

Question 14

Why does a magnet fix itself to a ferromagnetic material?

Answer 14

If you bring a magnet close to the material, the magnets B-field makes the magnetic moments in the material align and build up a magnetisation vector. This is equivalent to an induced closed current ring at the surface. This creates a force between the current and the induced magnetic field which pulls them together.

Question 15

What is Faraday’s Law?

Answer 15

The induced emf in a closed loop equals minus the rate of change of magnetic flux through the loop. ϵ =−N * dΦ/dt, where N is the number of turns in the coil.

Question 16

Describe how to use Faraday’s Law in the moving crossbar case.

Answer 16

• CIrcuit has no power source, bar moves at constant velocity v with a uniform B-field in negative z direction
• In time t bar moves distance vt
• Area of circuit loop crossbar completes is Lvt
• Magnetic flux increases by dΦ = BA = BLvt
• Therefore dΦ/dt = BLv, and e.m.f is counter clockwise: ϵ = -BLv

Question 17

How do you calculate the current in the crossbar problem?

Answer 17

Use Ohm’s law, ϵ = RI, where R is the resistance in the circuit, therefore I = BLv/R

Question 18

What is the equation for Ohmic heating?

Answer 18

RI^2 = (B^2*L^2*v^2)/R

Question 19

State Lenz’s law.

Answer 19

The effect of an induced emf is such that it opposes the changing flux which produced it.

Question 20

What is motional e.m.f?

Answer 20

Free charges inside the crossbar are forced to move at velocity v through magnetic field which creates an electric field as positive and negative charges accumulate on each end. The motional emf is the voltage between these two points.

Question 21

What happens to the magnetic flux at points in a cylindrical paramagnet when you drop a magnet through it with surface current Im in the φ-direction?

Answer 21

• At point z1 above the magnet the magnetic flux decreases, so dΦ/dt < 0
• At poijnt z2 below the magnet the magnetic flux increases, so dΦ/dt > 0

Question 22

What does Faraday’s law show in the magnet falling through cylinder problem?

Answer 22

That at z1 above the magnet an e.m.f is induced in the φ - direction and at z2 below the magnet an e.m.f is induced in the negative φ-direction.

Question 23

What does Ohm’s law show in the magnet falling through cylinder problem?

Answer 23

As the material has a finite resistance, and I = ϵ/R, a current is induced at points z1 and z2 in the φ-direction and negative φ-direction respectively.

Question 24

What does Ampere’s law show in the magnet falling through cylinder problem?

Answer 24

Induced wall current-loop produces a dipolar magnetic field centered at z1 and z2 in opposite directions.

Question 25

What is the force on the falling magnet?

Answer 25

F∼ IMφ(hat) X B = IM φ(hat) X B ϱ = − IMB ẑ B-field is negative so force is upwards and acts as a friction force slowing the magnet down.

Question 26

What is the equation for induced electric field in a stationary loop of wire around a solenoid?

Answer 26

int (E.dL) = -dΦ/dt

Question 27

What is flux linkage?

Answer 27

NΦ, where N is the number of turns in the coil.

Question 28

What does a resistor, capacitor and solenoid (coil) do?

Answer 28

• Resistor dissipates energy
• Capacitor stores energy in the electric field
• Soleoid stores energy in the magnetic field