Homework 3

Due in class Friday February 2^nd.

Note that as I set the problems from the Math book I assume that you are going to read the relevant sections of the Math book to help with the problems. I will cover what I can in class but I can't cover all the material. After all, that is what text books are for!

1) Math 5.76a. If U and V have continuous partial derivatives, proves that
∇(U + V) = ∇U + ∇V.

2) Math 6.67. Evaluate
∫^(4,2)_(1,1)[(x + y)dx + (y - x)dy]
along (a) the parabola y² = x, (b) a straight line, (c) straight lines from (1,1) to (1,2) and then to (4,2), (d) the curve x = 2t² + t + 1, y = t² + 1.

3) Math 6.68. Evaluate
∫[2x - y + 4)dx + (5y + 3x -6)dy]
round a closed triangular path in the x,y plane with vertices at (0,0, (3,0), (3,2) traversed in a counterclockwise direction. (Note that you will have to split the integral up into three integrals along straight lines).

4) G & P 1.3. The maximum electric field which can be supported by dry air at atmospheric pressure is about 10⁶ volts/m. What is the maximum potential difference to earth for a conductiong sphere of radius 10 cm in air? (Take the distance from earth to the sphere to be infinite; providing the actual distance is large compared to the radius of the sphere the error from this assumption will be small.)

5) G & P 1.13. A parallel beam of electrons of kinetic energy 2 keV is moving in a region of constant electrostatic potential and is incident on a metal grid which is held at a potential of 2 kV. S second grid behind the first one is parallel to it and is held at a potential V. If the electron beam direction is perpendicular to the grids, what is the smallest potential V required to prevent the beam from emerging? What value of V is requred if the beam is incident at 45°.

6) Find the electrostatic potential at all points in the bisector plane of the electric dipole in the figure below.

7) Consider a thin plastic rod bent into an arc of radius R and angle q'(see below). This rod carries a charge Q uniformly distributed along its length. Compute the electric field E at the origin.

Do this in the usual steps:

a) Draw a diagram showing what variables you will use, how you will cut the rod into elementary pieces, and what the dE from an elementary piece looks like.

b) Express dE algebraically in terms of the variables and the unit vectors i, j, and k.

c) Write the total field as an integral with appropriate boundary conditions and evaluate that integral.

d) Show that your answer makes physical sense.

8) For the same situation as in problem 7 find the potential at the origin by integrating the potential from the elementary pieces.