Maxwell's Equations in Matter Flashcards
What does an ideal electric dipole moment consist of?
Two equal but opposite charges a distance d apart, where qd= p, the dipole moment.
If we has a uniform cuboid aligned with cartesian axis which there are balancing charge densities +-ρ, what does the polarisation equal if we displace all the positive charges by a distance x in the positive x direction? Why does this happens?
Px = ρ*x. This polarisation happens because originally the positive and negative charges are on top of each other, but now the positive charges have been displaced so there is a charge imbalance on each face.
What is the equation for the surface charge density on the cuboid surface?
σ(P) = P(hat).n(hat)
What is the best way to look at the non-uniform polarisation?
Break the object into a series of small cuboids of length dx along the x axis, each with individually uniform polarisation.
Once we have broken an object up into cuboids, what do each of the cuboids have as polarisation?
Cuboid centred on x has polarisation +-Px(x), next one centred on x+dx has polarisation charge density +-Px(x+dx), and there is a charge excess of A(Px(x)-Px(x+dx)) on the interface between them, where A is the cross sectional area of the cuboid perpendicular to x.
What is the equation for polarisation charge density?
ρ(P) = (APx(x)-APx(x+dx))/Adx = -(Px(x+dx)-Px(x))/dx = -dPx/dx
In the limit for dx -> 0 and adding similar terms for y and z, what is the final equation for the polarisation charge density?
ρ(P) = -∇.P
What is an easy way to think of a sphere of radius a which has uniform P at all points?
Think of it as the superposition of 2 spheres of uniform volume charge densities +-ρ displaced from each other by x, such that P = ρx
For 2 spheres of charge density +/ρ centred at +- x/2, what is the equation for the field due to the polarisation charges at r inside the sphere?
E(r) = -P/3ε0
What is the equation for the polarisation of the sphere in terms of E0, the external field?
P = (3χ/(3+χ))ε0E0
What is the equation for the electric field inside the sphere?
E = E0-P/3ε0 = 3/(2+εr) *E0
What is the equation for the dipole moment outside of the spheres?
p = Qx = 4π/3 *a^3 *P
What happens if the polarisation in a material changes with time?
Charges must move, which creates polarisation currents which generate magnetic fields.
What is the equation for the polarisation current density J(P)?
J(P) = dP/dt (partial derivative)
What can magnetised materials be envisaged as?
As having distributions of small current loops.
What is the equation for magnetism if we split a material up into current loops?
M = m/V = IΔxΔy/ΔxΔyΔx = I/Δz, where each loop has area ΔxΔy, so magnetic moment is IΔxΔy
Which direction does the magnetisation point for a material with current loops in x-y plane?
z(hat) direction, so M = (I/Δz) z(hat)
What happens to the currents inside the block when the magnetisation is constant and uniform?
The current is constant so current from neighbouring loops cancel inside the material, leaving only current running along outside of block.
What is the equation for j(M), the general expression for the equivalent surface current density?
j(M) = M X n(hat)
What happens if the magnetisation varies with position inside the material?
The currents within the material will no longer cancel and a volume current density arises.
What is the equation for the residual current created by a varying magnetisation for two current loops centred at x and x+Δx?
ΔI = I*(x+Δx)-I(x) ~ dI/dx Δx
What is the equation for the volume magnetisation current J(M)?
J(M) = ∇ X M
In Gauss’s law, what do we separate the total charge density ρ into? Why do we do this?
ρ = ρ(f) + ρ(P) = ρ(f) - ∇.P, where f is free and p is polarisation. We separate this because the free charges are the ones we can manipulate, while the polarisation charges just come with the territory.
How can we rearrange the separated Gauss’s law to find the first of Maxwells equations in matter?
Put the divergence on the left hand side and multiply both sides by ε0, giving ∇.(ε0E + P) = ρ(f)
What is maxwell’s first equation in matter?
Define new quantity D, known as the displacement, where D = ε0E + P, so the first equation is ∇.D = ρ(f)
What is the integral form of maxwells first equation in matter?
closed integral of D.dS = integral of ρ(f) dV = Q(f) = free charge enclosed by the surface of integration on the left.
What is the equation for P for a given E? What can we do with this equation?
P = ε0χE, where χ is the electric susceptibility. Sub this equation into the equation for D, so D = ε0εrE, where εr = 1+χ = relative permittivity
What is χ (and therefore εr) in general dependent on?
The frequency, reflecting how fast a material can respond to a changing field.
How can we split the total current density J in Amperes law?
Split into free current density, a term due to magnetisation currents and a term due to changing polarisation: J = J(f)+J(M)+J(P) = J(f)+(∇XM) + dP/dt
What do we do with the split equation for J?
Sub it into amperes law, put the cur of the magnetisation on the left hand side and divide through by μ0.
What do we define the new vector field H as and where do we put it?
H = B/μ0 -M, so sub this into the new Amperes law to get the new maxwell equation in matter.
What is the second of maxwells equations in matter?
∇XH = J(f)+dD/dt
What happens to the other two unchanged maxwell equations?
They remain unchanged: only the first and the fourth change in matter as the others involve no source terms.
