Homework Problems

1. Assume that a charged wire, with constant charge density function occupies the interval [-1, 1] on the y-axis. Calculate the potential at the point P(1/2, 1/2) by setting up and solving a discrete model which expesses the scaled potential at this point. (Hint: Follow the ideas presented in this module and use Maple or Mathematica)

2. Assume that the charge density of the 1-D finite length wire follows the power law, i.e., it is proportional to . Execute the Trapezoidal and Simpson's programs and also use Maple to evaluate the potential at P(1/2, 1/2), for k = 0,1,...,10. Create a table of your results from the three methods. Are the results obtained from the Trapezoidal or Simpson's routine "closer" to those obtained by Maple/Mathematica? Give a reason.

3. Assume that a wire of infinite length and constant charge distribution resides on the y-axis. Use Maple/Mathematica to find the scaled potential at point P(1/2, 1/2). Explain mathematically why Maple/Mathematica produces such a result.

4. Assume that a thin charged disk, of radius R = 2 cm, lies on the yz-plane. The disk is uniformly charged with a constant surface charge distribution. Find the scaled potential at a point P (on the x-axis) which is 10 cm away from the center of the disk. Use a Computer Algebra System to perform analytic and numeric calculations of the integral which expresses the potential. (Hint: First, express the potential at the point P due to a ring of radius r and width dr).

5. Another force of similar nature to Coulomb's force is the gravitational force. Consider an 1-D rod of length 2 m, on the y-axis, centered at the origin with mass of 10 Kg. Assume that the mass distribution per unit length is of the form: . Find the gravitational potential at point P(1.50,3.50) with mass 0.001 Kg. Formulate the problem and express the potential as a 1-D integral. Use a Computer Algebra System to find the potential at the specified point. Make your own choice for the value of k.

6. Assume that a rigid rod of length L = 3 m rests on the x-axis. Calculate the moment of inertia about an axis perpendicular to the rod through one end. Let be the mass per unit length. Set up an appropriate integral to express the moment of inertia. Calculate the integral symbolically and/or numerically for the cases where:
   • is a constant;
   • ;
   • is proportional to your favorite function.