Physics 112 HW09
Due Monday, 29 September 2014
U3-VfEI03 or U3-VfEPC03. Consider the situation below. Two point charges +q and –q lie on the x axis. The +q charge is at x = +d/2 and the –q charge is at x = -d/2.
a) Sketch the electric field and equipotential lines in the xy plane.
Using both methods for finding V (and), derive expressions for
b) the electric potential V at points on the x axis for x > d/2, and
c) the electric potential V at points on the y axis. (Does your result make sense in light of your sketch in part a?)
You may assume that V → 0 as r → ±∞. For points on the x axis, which points away from the origin, and for points on the y axis, and points to the left. You may leave the integral in integral form, but isn’t it more fun to evaluate the integrals? Of course, you may use your favorite software to do so.
d) Let d = 1cm and q = 2 μC. A bead with mass 3 g and charge +1 nC is released from rest at the point x = d. Calculate its speed when it gets very far away from these two charges.
e) A very thin, straight (insulating) rod is placed on the y axis. Our bead is put at rest on this wire at y = d so that it is free to move without friction on the wire along the y-axis but cannot come off the wire (it is constrained to move on the y-axis only). Calculate its speed when it gets very far away from these two charges (trick question?).
U3-VfEdq00. A line charge extends along the x axis from –L/2 to L/2. It has a total charge +Q spread evenly over its length. Determine an expression for the voltage (electric potential) for all points on the x axis such that x > L/2. You may leave your expression in the form of an integral, but it may only contain the terms Q, L, x, the variable of integration, and fundamental constants.
U3-VfEdq01A. An insulating semicircular wire of radius R = 2cm has a charge of 5 nC painted uniformly along its top half, and a charge –5 nC painted uniformly along its bottom half. Calculate the electric potential at the center (point P) assuming that V → 0 as we get infinitely far away from the semicircle. (Trick question?)
U3-VfEdq02. (Wolfson, Ch. 22 Problem 77) A line charge extends along the x axis from –L/2 to L/2. Its linear charge density is λ(x) = λo (x/L)2, where λo is a constant. Find an expression for the electric potential on the x axis for x > L/2. You may leave your expression in the form of a definite integral, but it may only contain the terms o, L, x, the variable of integration, and fundamental constants.
U3-VfEI04. A 2.0-cm radius metal sphere carries 75 nC and is surrounded by a concentric spherical conducting shell of radius 10 cm carrying -75 nC.
a) Find the voltage as a function of distance from the middle of the spheres everywhere, assuming that V → 0 as r → ∞.
b) What is the potential difference between the two spheres?
c) Repeat a) and b) if instead the shell carries a net +150 nC.
U3-VfEI08. (Wolfson, Ch. 22 Problem 59) (a) Find the electric potential as a function of position on the x and y axes if the electric field is given by , where Eo is 150 V/m (150 N/C) and andare unit vectors in the +x and +y directions, respectively. Take the zero of electric potential at the origin. (b) Find the potential difference from the point (x, y) = (2.0 m, 1.0 m) to the point (x, y) = (3.5 m, -1.5 m)
(over)