Intro & Maxwells Equations Flashcards
What is the equation for the ?
F = q(E+v X B)
What is Coulombs law?
E(r) = q/4πε0r^2 r(hat)
What is the equation for the Coulomb potential?
ψ(r) = q/4πε0r
What is the equation for Gauss’s law?
ф = closed integral over S of E.dS = Q/ε0 = 1/ε0 *integral over V of ρ dV
What is the equation for the Biot-Savart law?
B(r) = μ0/4π *q *(v X r(hat))/r^2
What is the equation for the Amperes law?
B(r) = μ0I/2πr, or the closed integral over C of B.dl = μ0* integral over S of J.dS
What is the equation for the Faraday-Lenz law of induction?
closed integral over C of E.dl = -dф/dt
How do we manipulate Gauss’s law to get the first maxwell equation?
Use Gauss’s mathermatical theorem: closed integral over S of E.dS = integral over V of ∇.E dV -> combine this with Gauss’s law and put all on one side and in one integral. Shrink integral to tiny region.
What is Maxwells first equation?
∇.E = ρ/ε0
What is the solenoidal condition on B?
Magnetic flux lines are always closed loops, so closed integral over S of B.dS = 0
How do we get Maxwells second equation?
Follow same procedure as the first equation to get answer.
What is Maxwells second equation?
∇.B = 0
How can we use Faraday_Lenz law of induction with Stokes’s theorem?
Stokes theorem: closed integral over C of E.dl = integral over S of ∇XE.dS, so sub this into Faraday-Lenz law
How do we manipulate the new equation from Stokes theorem?
Same way as manipulating Gauss’s law: put all on one side and reduce integral to vanish.
What is Maxwell’s third equation?
∇XE = -dB/dt -> differential is partial
How can we prove the consistency between Maxwell’s third and fourth equations?
Take divergence of both sides of third equation and find that dB/dt = 0, confirming ∇XB = 0
What do we do to Ampere’s law to start deriving Maxwell’s fourth equation?
Same as third: use stokes’s theorem and sub in, then reduce integral to zero.
What is the differential version of Maxwells fourth equation?
∇XB = μ0*J
Why doesn’t the differential version of Maxwells fourth equation work?
Take divergence of both sides, and find that ∇.J = 0, which is wrong, as J is the flux density of electric chargeso ∇.J is the amount of stuff produced per unit volume.
How do we derive the continuity equation for electric charge?
Consider total charge in arbitrary volume: Q = integral over V of ρ dV, so dQ/dt = d/dt*integral over V of ρ dV = - closed integral over S of J.dS
What is the continuity equation for electric charge?
dρ/dt + ∇.J = 0
How can we modify the continuity equation with Gauss’s law and charge density?
Use Gauss’s law to sub the charge density in the continuity equation in terms of divergence of E, since ρ = ε0*∇.E
What is the ‘subbed in’ version of the continuity equation?
ε0∇.dE/dt + ∇.J = 0, therefore ∇.(J+ε0dE/dt) = 0. Can use this modified version of J to sub into the differential form of Maxwells fourth equation.
What is Maxwells fourth equation?
∇ X B = μ0(J+ε0dE/dt)
What does the extra term added onto J in Maxwells fourth equation mean?
It is the displacement current density
