Chapter 4.4, Problem 4.19P

Problem 2.28 Use Eq. 2.29 to calculate the potential inside a uniformly charged solid sphere of radius R and total charge q. Compare your answer to Prob. 2.21.

Consider an infinitely long wire of charge carrying a positive charge density of A. The electric field due to λ this line of charge is given by E= 2kef= -, where is a unit vector directed radially outward Σπερμ from the infinitely long wire of charge. Hint #3 a. Letting the voltage be zero at some reference distance (V(ro) = 0), calculate the voltage due to this infinite line of charge at some distance r from the line of charge. Give your answer in terms of given quantities (A,ro,r) and physical constants (ke or Eo). Use underscore ("_") for subscripts and spell out Greek letters. Hint for V(r) calculation 3 V(r) = b. There is a reason we are not setting V(r → ∞o) = 0 as we normally do (in fact, in general, whenever you have an infinite charge distribution, this "universal reference" does not work; you need a localized charge distribution for this reference to work). Which of the following best describes what happens to potential as roo? (That is, what is V(ro), with our current…

The space between the plates of a parallel-plate capacitor (Fig. 4.24) is filled with two slabs of linear dielectric material. Each slab has thickness a, so the total distance between the plates is 2a. Slab-1 has dielectric constant of 2 and slab-2 has a dielectric constant of 1.5. With the area of each of the top and bottom conducting plates is much greater than a?, we can assume the the surface charge densities +o and -o on the top and bottom plates is uniform. (a) Find the electric displacement D in each slab. (b) Find the electric field E in each slab. (c) Find the potential difference between the plates. (d) Find the locations and amounts of all bound charge. (e) Based on the values of bound charge, recalculate E and verify your answer from (b). (f) How do your results relate to the formula for the addition of two series capacitors?

For problem 4 part b in square centimeters using inner and outer radii of the spherical capacitor of a = 2.00 cm and b = 1.05 a, respectively. (Answer in 5 sig. figs.)

Problem 1. Prove that for a vacuum-dielectric interface at glancing incidence ri→-1 (see Fig. 4.49 from textbook, also on slide 7 in Lecture 4). In the same figure, if a is the angle that the curve r(0.) makes with the vertical at 0; = 90°, then: Vn2 – 1 tana, 2 1.0 0.5 Op -0.5 56.3° -1.0 30 60 90 0; (degrees) Figure 4.49 The amplitude coefficients of reflection and transmission as a function of incident angle. These correspond to external reflection n; > n; at an air-glass interface (n = 1.5). Amplitude coefficients of

1.13 Two infinite grounded parallel conducting planes are separated by a distance d. A point charge q is placed between the planes. Use the reciprocation theorem of Green to prove that the total induced charge on one of the planes is equal to (-q) times the fractional perpendicular distance of the point charge from the other plane. (Hint: As your comparison electrostatic problem with the same surfaces choose one whose charge densities and potential are known and simple.)

Problem 3.18 The potential at the surface of a sphere (radius R) is given by Vo = k cos 30, where k is a constant. Find the potential inside and outside the sphere, as well as the surface charge density o (0) on the sphere. (Assume there's no charge inside or outside the sphere.)

Figure 1.52 shows a spherical shell of charge, of radius a and surface density σ, from which a small circular piece of radius b << a has been removed. What is the direction and magnitude of the field at the midpoint of the aperture? Solve this exercise in three ways: a) direct integration, b) by superposition, and c) using the relationship for a force on a small patch.

[ Q.3 A capacitor has orthogonal plates of length a and width b. The distance between the plates is h<< a, b. The capacitor has a charge Q. The space between the plates is initially vacuum and we insert slowly a slab of dielectric material as of dielectric constant & as shown in figure. What is the force that is exerted on the slab when it has entered a distance x inside the capacitor? a