Homework 4

Due in class Friday February 9^th

1) Math 5.76b. If A and B have continuous partial derivatives, proves that
∇•(A + B) = ∇•A + ∇•B.

2) Math 5.77b. If φ = xy + yz + zx and A=x²yi +y²zj + z²xk find
φ∇•A.

3) Math 5.85. Find the equations of the (a) tangent plane and (b) normal line to the surface x² + y² = 4z at (2,-4,5).

4) G & P 1.4. Thie figure below shows a cross-section of the cylindrical high-voltage terminal of a Van Der Graaf generator, surrounded by an 'intershield' and pressure vessel, both of which are also cylindrical. The gas in the pressure vessel breaks down in electric fields greater than 1.6x10⁷ volts/m. If the radii of the terminal, intershield, and pressure vessel are 1.5m 2.5m, and 4m respectively, what is the highest potential difference that can be maintained between the terminal and the pressure vessel?
(Hint: the intershield must be maintained at a potential such that the breakdown is about to occur on its own outer surface as well as the on the surface of the terminal.)
Note that this breakdown field is a lot greater than that of air. In my experience these things are filled with sulphur hexafluoride under several atmospheres pressure. You really don't want a leak!

5) G & P 1.8. Two large parallel plates of area A are a distance d apart and are maintained at potentials 0 and V. A third similar plate, carrying a charge q, is isolated from the other plates and placed midway between the two. What is the potential on this plate?

6) G & P 1.12. The field emission microscope consists of a fine hemispherical tungsten point at the centre of an evacuated sphere, the inside surface of which is coated with a fluorescent screen. If the fluorescent screen is maintained at a positive potential with respect to the point, there is a large electric field at the point. When the magnitude of this field reaches about 10⁸ V/m, electrons are drawn from the point even at room temperature. This process is called field emission. The electrons leaving the point follow the field lines an a magified image of objects on the point appears on the screen. In a typical microscope the radius of curvature of the point and the screen are 10 nm and 10 cm respectively. What is the magnification of the microscope? What is the minimum potential difference needed to cause field emission?

Hint. Draw a good diagram and assume that conditions in half a sphere are nearly the same as if the rest of the sphere were there. This will be true well away from the end of the hemisphere and can be made almost perfectly true by adding fine wire grids at the edges.

7) A thin, non-conducting, plastic rod is bent into the form of a nearly complete circle with a radius of 50 cm. There is a 2 cm gap between the ends. A positive charge of 1 C is spread uniformly over the length of the rod. What is the magnitude and direction of the electric field at the centre of the circle? (Hint. Symmetry will help but you also need to remember that the principle of superposition works just as well for subtracting fields as for adding them. You should also note that you have found the answer to a related problem in the last homework that should be a great help in doing this.)

8) Two infinite plane sheets of charge, one of density s = 6 C/m2 and one of density s = -4 C/m2 are located 10 cm apart, parallel to one another. Describe the electric field everywhere in this system.

9) Some time ago we found the Electric Field from a finite length rod of charge in a couple of special directions by integrating the field from little lengths of the rod. Now write an integral giving the electric potential at any point in space. You should probably work in cylindrical polar coordinates and do not need to evaluate the integral.