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Homework 2 To be submitted till 15/11/11 1 Coulomb's Law: 1. Find the electric eld a distance z above one end of a straight line segment of length L in the following gure: , which carries a uniform line charge λ. Check that your formula is consistent with what you would expect for the case z L. (Grit, problem 2.3) 2. Find the electric eld a distance z above the center of a square loop (side a) carrying uniform line charge λ, in the following gure: . (Grit, problem 2.4) 3. Find the electric eld a distance z from the center of a spherical surface of radius R in the following gure: , which carries a uniform charge density σ . Treat the case z < R (inside) as well as z > R (outside). Express your answers in terms of the total charge q on the sphere. Do not use the Gauss law! (Grit, problem 2.7) 4. Use your result in Prob. 3 to nd the eld inside and outside a sphere of radius R, which carries a uniform volume charge density ρ. Express your answers in terms of the total charge of the sphere, q . Draw a graph of |E| as a function of the distance from the center. (Grit, problem 2.8) 2 Gauss Law: 1. Find the electric eld inside a sphere which carries a charge density proportional to the distance from the origin, ρ = kr, for some constant k. (Grit, problem 2.14) 2. A hollow spherical shell carries charge density ρ = k r2 in the region a ≤ r ≤ b in the following gure: . Find the electric eld in the three regions: (a) r < a (b) a < r < b (c) r > b Plot |E| as a function of r. (Grit, problem 2.15) 3. A long coaxial cable, in the following gure: , carries a uniform volume charge density ρ on the inner cylinder (radius a), and a uniform surface charge density on the outer cylindrical shell (radius b). This surface charge is negative and of just the right magnitude so that the cable as a whole is electrically neutral. Find the electric eld in each of the three regions: (a) inside the inner cylinder r < a (b) between the cylinders a < r < b (c) outside the cable r > b Plot |E| as a function r (cylindrical coordinate). (Grit, problem 2.16) 4. Two spheres, each of radius R and carrying uniform charge densities +ρ and −ρ, respectively, are placed so . Call the vector from the that they partially overlap, as illustrated in the following gure: positive center to the negative center d. Show that the eld in the region of overlap is constant, and nd its value. (Grit, problem 2.18) 3 Electric Potential: 1. For the charge conguration of Prob. 2.2, nd the potential at the center, using innity as your reference point. (Grit, problem 2.23) 2. For the conguration of Prob. 2.3, nd the potential dierence between a point on the axis and a point on the outer cylinder. (Grit, problem 2.24) 3. Two point charges ±Q are located at x = ∓ d2 , respectively. This system is called an dened as p = Qd, where d is a vector, pointing from −Q to +Q. electric dipole and (a) Find the electric potential in any point r in space outside the dipole. Hint: Use vectors R± and angle θ in the following gure: . (b) Draw surfaces that have an equal electric potential. (c) Simplify the electric potential in (a) assuming d r. 4. A straight line segment of length d located in the xy plane, as illustrated in the following gure: carries a uniform line charge λ. (a) Find the electric potential in any point r in space outside the line segment. (b) Simplify the electric potential in (a) assuming y x, z, d. 5. A conical surface (an empty ice-cream cone) carries a uniform surface charge σ . The height of the cone is h, as is the radius of the top. Find the potential dierence between points a (the vertex) and b (the center of the top). (Grit, problem 2.26) ,