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Physics 318 Electromagnetism R.G. Palmer 2/9/09 Test 1 1. Consider a three-dimensional vector field V (x) such that ∇·V =s (1) ∇×V =c (2) with, of course ∇ · c = 0, and s and c are bounded. Show that if V = −∇φ + ∇ × A where (r ≡ |x − x0 |) Z s(x0 ) 3 0 1 dx 4π r Z 1 c(x0 ) 3 0 A(x) = dx 4π r φ(x) = then V satisfies eq (1) and (2). You can assume ∇2 1r = −4πδ(r), and any formulas and theorems from Jackson’s endnotes. 2. A long right circular cylinder (a rod) of dielectric constant κ of radius a has its axis coincident with the z and its ends at z = −L/2 and z = L/2 with L a. There is a previously uniform electric field E 0 along the cylinder (i.e. E 0 = E0 k̂). It may not be what you first think. a. Show that the bound volume-charge ρb is zero. b. Find the fields E, D, and P at the center (x = y = z = 0). c. Find the field E, D, and P just inside and outside the end of the cylinder along the axis (x = y = 0, z = 21 L± . d. Make two sketches for E and D, and add the bound charges (+, −) on at least one of your sketches. 3. A toroidal coil consists of a circular ring, or “donut,” around which a long wire is wrapped. The winding is uniform and tight enough so that each turn can be considered a closed loop. The coil has radius a, N turns per unit length, and the current I. Find the magnetic field B at all points, both inside and outside the coil. You must show the direction of the field as well as the magnitude.