Week 2 Flashcards
Charge from charge density
Q = ∫dq = ∫ρ(r,t)dV
Current from current density
I = ∫dI = ∫j(r,t) ⋅ dS
Derive the continuity equation
Conservation of charge implies that rate of loss of charge is equal to the current, so Q̇ = -I
Using charge density to total charge equivalence, and the divergence theorem for current density, we get
∫ᵥρ̇dV = Q̇ = -∫ₛj ⋅ dS = -∫ᵥ∇ ⋅ jdV
therefore,
∫ᵥ{ρ̇ + ∇ ⋅ j} dV = 0
Must hold over any volume V, so integrand must be zero, giving final continuity equation
ρ̇ + ∇ ⋅ j = 0
Units of electric field, magnetic field, electric flux, and magnetic flux?
Electric field: E(R,t) | Vm⁻¹
Magnetic field: B(r,t) | T
Electric flux: ΦE = ∫ₛE ⋅ dS | Vm
Magnetic flux: Φᴮ = ∫ₛB ⋅ dS | Wb = Tm² = Vs
Lorentz force law?
f = E + v ⨯ B
Electromagnetic energy density
u = 1/2 ε₀|E|² + 1/2μ₀ |B|²
units Jm^-3
Total electromagnetic energy in a volume V
Uₜₒₜ = ∫ᵥu dV
unit J
Electromotive force along line
ε = ∫E ⋅ dl
unit V
Gauss’ law
(basic integral form)
Flux of electric field through closed surface S is proportional to the amount of charge enclosed by that surface.
∫ₛE ⋅ dS = ΦE = Q/ε₀ = 1/ε₀ ∫ᵥ ρdV
Gauss’ law
(Volume integral)
∫ᵥ∇ ⋅ EdV = 1/ε₀ ∫ᵥ ρdV
∫ᵥ(∇ ⋅ E - ρ/ε₀) dV = 0
Gauss’ law
(differential form)
∇ ⋅ E = 1/ε₀ ρ
No magnetic monopoles
(integral form)
The flux of magnetic field through any closed surface is zero.
∫ₛB ⋅ dS = Φᴮ = 0
No magnetic monopoles
(differential form)
∇ ⋅ B = 0
Proves no sources or sinks.
Faraday-Lenz law
(integral form)
The electromotive force (EMF) induced in a closed loop is equal to the rate of change of magnetic flux linked in the loop.
The EMF is generated in a direction such that the current flow will annul the change in flux.
ε = -d/dt Φᴮ
recall ε = ∫E ⋅ dl
∫ᴸE ⋅ dl = ∫ₛ ∇ ⨯ E ⋅ dS = -∫ₛḂ ⋅ dS
Faraday-Lenz law
(differential form)
∇ ⨯ E + Ḃ = 0
