Magnetism

Question 1

force on a charged particle due to a magnetic field

Answer 1

F = q (v x B)
q: charge
v: velocity
B: magnetic field
x: cross product

Question 2

Lorentz force equation

Answer 2

F = q[ E + (v x B) ]
q: charge
E: E-field
B: magnetic field
v: velocity
x: cross product

Question 3

prove that magnetic fields never do any work on charged particles

Answer 3

W = ∫ F⋅dl
F = q (v x B) and dl = vdt

hence W = ∫q (v x B) ⋅ v dt
v x B is perpendicular to v.
∴(v x B) ⋅ v = 0
so W = 0

Question 4

force on a current carrying wire equation

Answer 4

F = L (I x B)
L: wire length in B-field
B: magnetic field
I: current
x: cross product

Question 5

Biot-Savart law equation

Answer 5

dB = µ₀/4π * (𝑰dl x 1/r² r̂)
dB: magnetic field
𝑰: Current
r: distance from the wire
dl: direction of current flow
r̂: radial direction

Question 6

Gauss’s law for magnetic fields

Answer 6

since magnetic field lines form closed loops, the flux entering an leaving any Gaussian surface will be the same

hence: ∮B ⋅ dA = 0

Question 7

Ampere’s law (aka B-field circulation)

Answer 7

∮B⋅dl =µ₀*I = µ₀∫ j⋅dA
dl: length element of a closed loop (Amperian loop)
I: the current enclosed within the loop
j: current density vector
dA: area element of the area bound by the curve dl

Question 8

important solenoid facts

Answer 8

B inside is uniform
B outside is 0

Question 9

magnetic moment equation

Answer 9

µ = AI
µ: magnetic moment
A: loop area
I: current

this is for a loop of current

Question 10

equations about a magnetic dipole inside a uniform magnetic field

Answer 10

T =µ x B
T: torque (tau symbol)
µ: magnetic moment
B: field strength of the uniform field

U = -µ⋅B
U: potential energy
µ: magnetic moment
B: field strength of the uniform field