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test1-S10

# test1-S10 - K and applying a Lorentz transformation...

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Physics 318 R.G. Palmer Electromagnetism 2/22/10 Test 1 1. Prove the following identities. You should just assume the fundamental theorems for grad, div, and curl (Stokes’s theorem). a. I C u v · d l = - I C v u · d l b. Z V ∇ × A d 3 x = I S n × A da c. Z S n × ∇ ψ da = I C ψ d l 2. a. In its own rest frame (in a frame K 0 ), the potential of a point charge is Φ 0 = q 4 πε 0 r 0 A 0 = 0 Find the 4-vector potential, A α = (Φ /c, A ), of a moving charge by placing the charge at the origin of a frame K 0 moving at velocity v in the ˆx direction with respect to frame K and applying a Lorentz transformation. Show that the components are given by the formulas Φ = γq 4 πε 0 p γ 2 ( x - vt ) 2 + y 2 + z 2 A x = v c 2 Φ A y = A z = 0 b. In its own rest frame (in a frame K 0 ), the potential of a point electric dipole p is Φ 0 = p · r 0 4 πε 0 r 0 3 A 0 = 0 Find the 4-vector potential, A α = (Φ /c, A ), of a moving dipole by placing the dipole at the origin of a frame K 0 moving at velocity v in the ˆx direction with respect to frame
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Unformatted text preview: K and applying a Lorentz transformation. 3. [Jackson 4.10] Two concentric conducting spheres of inner and outer radii a and b , respectively, carry charges ± Q . The empty space between the spheres is half-ﬁlled by a hemi-spherical shell of dielectric (of dielectric constant ε/ε ), as shown in the ﬁgure. a. Find the electric ﬁeld everywhere between the spheres. b. Calculate the surface-charge distribution on the inner sphere. c. Calculate the polarization-charge density induced on the surface of the dielectric r = a (i.e. σ b ). Hints: (1) Guess that the E is radial. (2) The Gauss’s law is H S E · ˆn da = q inside /ε and also H S D · ˆn da = q inside f /ε ....
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