Homework 1

Due in class Friday January 19th.

1) Three point charges are placed at the vertices of an equilateral triangle with sides one metre long.The charges are all positive and each of magnitude 1µC. What is the force on each charge? What is the magnitude of the electric field (a) at the centre of the triangle and (b) at the mid-point of one side of the triangle.

2) Three charges are placed along a straight line at positions x = -0.1m, x = 0m, and x = +0.1m. If the charges have values 1µC, -0.2µC, and q, what must be the value of q in order for the net force on it to be zero?

3) If A = 2i + j + k, B = i - 2j + 2k, and C = 3i -4j + 2k, find the projection of A+C in the direction of B.

4) If A x B = A x C, does B = C necessarily?
Hint: If the answer is yes then we need a proof. If the answer is no then the explanation should include a counter-example.

5) A particle moves along the space curve r = exp(-t) (i cos(t) + j sin(t) + k).
a) Sketch the curve of r.
b) Find the magnitude and direction of the velocity at any time t.
c) Find the magnitude and direction of the acceleration at any time t.

6) If A is a differentiable function of u and |A(u)|=1, prove that dA/du is perpendicular to A.
Hint: Remember that |A|² = A•A.

7). Using the diagram below find the magnitudes of the charges on the two spheres. The spheres are metal coated pith balls and are small enough that you can assume that the charge spreads evenly over their surfaces so that they act like point charges. Each pith ball weighs 5g and the strings weigh so little that you can ignore their weight. You should assume that the balls were initially in contact so that they each carry the same charge. They were then released and allowed to come to equilibrium when they hang as you see them in the figure.

8) In class we found the field in the bisector plane near a short rod of charge. Use the same method to find the field outside the rod along the line of the charge. That is, find the field along the z axis in the figure below.