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1
Q

What is the equation for Coulomb’s Law?

A

F12 = (q1q2)/(4πϵ0)*(r2 − r1)/(|r2 − r1|^3)

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2
Q

What is the value of the permittivity of free space and what are its units?

A

8.8*10^-12 C^2 N^-1 m^-2

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3
Q

What is the equation for an electric field around a point charge q?

A

E(r) = q/(4πϵ0)*(1/r^2) (same as coulombs law, but with only one charge rather than two)

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4
Q

What is the equation for the electric dipole moment?

A

p = qd q = charge d = distance between charges p = moment [Units: Cm]

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5
Q

How do you work out the electric flux through a flat surface?

A

flux = electric field * area of surface * cos(angle)

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6
Q

How do you find the electric flux through an arbitrary surface?

A

Surface integral of the electric field * the change in area/surface (electric field is constant on surface so can take out of integral).

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7
Q

What is the surface integral of E*dS of a sphere centered on a charge q?

A

q/ϵ0

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8
Q

What is the equation for potential energy associated with the coulomb interaction of two charges?

A

(q1q2)/(4πϵ0*r)

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9
Q

What is the equation for potential energy of a point charge at a point in space?

A

q/(4πϵ0(r-r’))

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10
Q

How do you calculate a potential?

A

You find the integral between infinity and a point a within the electric field of E*dl, where E is the electric field and l is the distance it moves.

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11
Q

How do you calculate potential gradients?

A

-vV = -dV/dx i - dV/dy j - dV/dz k = E where V is the potential,v is the gradient and E is the electric field.

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12
Q

What is the equation for capacitance?

A

C = Q/V, where Q is the charge and V is the potential difference between the plates.

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13
Q

What is the equation for capacitance of two parallel plates? How do you derive this?

A

C = ϵ0*A/d Electric between the plates is -σ/ϵ0, where σ is the surface charge density (Q/A). Thus, E = Q/(ϵ0*A). For uniform potential difference, V = Ed, so V = (Q*d)/(ϵ0*A). Rearrange for capacitance.

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14
Q

How do you derive the capacitance of a cylindrical capacitor?

A

Use the potential of an infinite line of charge and set the two radii to the radii of the inner and outer cylinder. Then use lambda = Q/L

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15
Q

How do you derive the capacitance of a spherical capacitor?

A

Use the potential of a sphere centered on a charge q but change 1/r for 1/r2 - 1/r1, where r2 and r1 are the radii of the respective spheres. Then use the charge stored on each sphere as q = σ4πr^2, where σ is the surface charge density.

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16
Q

What is the self-capacitance of a single spherical plate of radius r?

A

C = 4πϵ0r

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17
Q

How do you add the capacitance of capacitors in series/parallel?

A

Series: 1/C = 1/C1 + 1/C2 + … Parallel: C = C1 + C2 + …

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18
Q

What are the 3 main equations for energy stored in a capacitor?

A

1/2 * Q/V 1/2 * C*V^2 1/2 * Q^2/C

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19
Q

What is the general equation for energy density?

A

energy density = energy/volume = u

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20
Q

What is the equation for relative permittivity?

A

ϵr = C/C0 where C0 is the capacitance when separated by a vacuum and C is when separated by the dielectric

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21
Q

What is the equation for permittivity of a material?

A

ϵ = ϵr*ϵ0

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22
Q

What is the dielectric breakdown?

A

When dielectrics become conductors as the electrons become unbound.

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23
Q

What is the equation for the current density?

A

J = I/A = nqv, where n is the number density of free charges and v is the mean velocity of the charges.

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24
Q

How do you find the mean speed of electrons in a wire?

A

From start point to point P, electron travels v*t, so volume of region passing P is v*t*A, where A is the cross sectional area. Number of free charges in volume is n*v*t*A where n is the number density. Charge passing P in t is dQ = q*n*v*t*A. Then I = dQ/dt, so find the derivative and rearrange for v.

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25
What is resistivity?
Determines how strong a current can flow for a given imposed electric field. Conductor - low resistivity, Insulator - high resistivity.
26
What is the reciprocal of resistivity?
Electrical conductivity
27
What is the equation for electrical resistance for any conductor?
ρL/A
28
State ohms law.
R = V/I
29
What is the total resistance of resistors in series/parallel?
Series - R = R1 + R2 + ... Parallel - 1/R = 1/R1 + 1/R2 + ...
30
What are the 3 main equations for power?
P = V\*I P = R\*I^2 P = V^2 / R
31
What is e.m.f?
The emf of a battery is the voltage across the terminals if there are no losses in the battery.
32
What us the equation for voltage including internal resistance?
V = ϵ - rI where ϵ is the e.m.f and r is the internal resistance.
33
How do you use maxwell loops?
Choose a direction of the loop. Find the voltages across each resistor and the total e.m.f in the loop and then solve. Also use Kirchoff's Laws about junctions.
34
What is the time constant and what is its equation?
The time it takes a capacitor to be charged. The equation is τ = RC
35
What is the equation for magnetic flux?
ΦB = surface integral of B\*dS
36
What is the solenoidal condition?
Flux entering a surface must also exit that surface - surface integral of B\*dS = 0
37
What is the equation for the force on a charge in a magnetic field?
F = qv X B, where v is the perpendicular velocity and B is the magnetic field strength.
38
What is the larmor radius and how do you get the equation?
Make the force on a charge in a magnetic field equal centripetal acceleration, (m\*v^2)/r.
39
What is the Lorentz force?
qv X B where X is the cross product.
40
What is the force on a section dL of wire?
dF = IdL X B
41
What is the force on a crossbar on rails?
F = ILB
42
What is the equation for the torque on a current loop?
μ x B where μ= IA sin(ϕ), or NIA sin(ϕ) , where N is the number of loops
43
What is the permeability of a vacuum and what is its value?
μ0 = 4π\*10^−7 H m^−1
44
State the law of Biot-Savert.
B = (μ0/4π)\*q\*((v(t) X (r-r'(t)))/(|r-r'(t)|^3))
45
What are all the different forces included when two charges move next to each other?
Electrostatic force and the lorentz force after working out the magnetic field (law of biot-savert).
46
What is the speed of light in terms of μ0 and ϵ0?
1/sqrt(μ0\*ϵ0)
47
What is the value of ϵ0?
8.85\*10^-12
48
What is the equation for the magnetic field of a straight wire?
B= (μ0\*I)/(2π\*) where ϱ is the distance from the centre of the wire.
49
What is Ampere's Law?
It states that the line integral of a magnetic field around a closed loop is equal to μ0\*I(encl).
50
How do you use Amperes law to find the magnetic field of things?
Similar to Gaussian surface - pick a line integral around the wire with current in, calculate the integral, make it equal to μ0\*I(encl) and then do it for different conditions (when I(encl) changes.
51
What 3 things can we assume about a solenoid?
-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
52
What are the 4 steps to calculating the magnetic field around a solenoid?
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
53
What is the magnetic dipole moment μ defined as?
μ = IA n(hat)
54
What does the current in μ generate itself in a dipole?
It generates a magnetic field which is parallel to μ (the axis of the current loop)
55
What does the torque in a dipole do?
It acts to align μ and B0, where μ is the normal vector to the current loop multiplied by IA.
56
What can atoms be said to behave like? What does this create?
Magnetic dipoles. If there are many small dipoles (atoms), it creates a surface current, and the current inside the material is zero.
57
What happens if you place a material into an external magnetic field B0?
B0 induces a μ and forces the atoms in the material to align to B0, and the atoms hen create their own magnetic field.
58
When a material is exposed to an external magnetic field B0, what happens to the magnetic field inside the material?
It becomes B = μr B0, where μr is the relative permeability which is a property of the material.
59
What are the 4 types of magnetic materials?
-Diamagnetic -Paramagnetic -Ferromagnetic -Hard Ferromagnetic
60
What is a diamagnetic material?
μr \< 1, so the B-field inside the material is reduced.
61
What is a paramagnetic material?
μr \> 1, so the B-field inside the material is increased.
62
What is a ferromagnetic material?
μr \>\> 1, so the B-field inside the material is greatly increased.
63
What is a hard ferromagnetic material?
Permanent magnets which maintain a magnetic field even when there is no external magnetic field. μr not defined.
64
Why does a magnet fix itself to a ferromagnetic material?
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.
65
What is Faraday's Law?
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.
66
Describe how to use Faraday's Law in the moving crossbar case.
-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
67
How do you calculate the current in the crossbar problem?
Use Ohm's law, ϵ = RI, where R is the resistance in the circuit, therefore I = BLv/R
68
What is the equation for Ohmic heating?
RI^2 = (B^2\*L^2\*v^2)/R
69
State Lenz's law.
The effect of an induced emf is such that it opposes the changing flux which produced it.
70
What is motional e.m.f?
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.
71
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?
-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
72
What does Faraday's law show in the magnet falling through cylinder problem?
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.
73
What does Ohm's law show in the magnet falling through cylinder problem?
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.
74
What does Ampere's law show in the magnet falling through cylinder problem?
Induced wall current-loop produces a dipolar magnetic field centered at z1 and z2 in opposite directions.
75
What is the force on the falling magnet?
F∼ IMφ X B = IM φ X B ϱ = − IMB ẑ B-field is negative so force is upwards and acts as a friction force slowing the magnet down.
76
What is the equation for induced electric field in a stationary loop of wire around a solenoid?
int (E.dL) = -dΦ/dt
77
What is flux linkage?
NΦ, where N is the number of turns in the coil.
78
What is the equation for self inductance L? What are its units? What is its circuit diagram symbol?
NΦ/I [Henry] It's circuit diagram symbol is a coil.
79
What is the self-induced e.m.f in an inductor? What does this mean?
ϵ = -L dI/dt This means there is a voltage drop across the inductor.
80
What is the equation for energy input/stored in L?
u = 1/2 \* L\*I^2
81
What is the equation for energy density of L?
B^2/2μ0
82
What happens to inductance when inductors are in series/parallel?
Same relationship as resistors.
83
How do transformers work?
Two inductors with same cross-section but different number of windings. Using Faraday's Law, ϵs/Ns = ϵp/Np, therefore if Ns\> Np, it steps the e.m.f up, and if Ns \< Np it steps the voltage down.
84
How can an AC voltage be represented in complex form?
V = V0\*e^(iωt) = V0(cos(ωt) + i\*sin(ωt))
85
How can you represent a phase difference in complex form?
V1 = V0\*e^(iωt) V2 = V0\*e^(iωt + iϕ) = V1\*e^iϕ = rotation by ϕ in argand diagram
86
What is the equation for the voltage of a DC and AC supply?
-DC - V = RI -AC - V = LI, where L is the complex impedance or the impedance.
87
What is the complex impedance of a resistor?
L = R
88
What is the complex impedance of a capacitor? How do you derive this?
L = 1/jωC Use Q = CV and I = dQ/dt, then introduce complex form and rearrange (including cos and sin and stuff)
89
What is the complex impedance of an inductor? How do you derive this?
L = jωL Use V = L dI/dt, and V(t) = V0 cos(ωt), and find the current I(t). Then introduce complex notation and rearrange.
90
How do you find the true value of something if it has a complex number in it?
Find the magnitude of the 'vectors' (so j or i is squared and therefore removed)
91
In capacitors and inductors, what is the phase difference between V(t) and I(t)?
pi/2
92
How much power is dissipated in inductors and capacitors with an AC current flowing through them?
None.
93
What is the equation for rms V and rms I?
Irms = I0/sqrt(2) Vrms = V0/sqrt(2)
94
What is the equation for power in an LRC circuit?
P = 1/2 \* V0 \* I0 \* cos(ϕ) = Vrms \* Irms \* cos(ϕ)

Electricity and Magnetism (8 decks)

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