Practice Problems: Conductors in Electrostatic Equilibrium Solutions
1. (easy) A spherical hunk of solid metal (R = 0.1 m) is charged with trillions and trillions of excess electrons. Electrostatic equilibrium is established so that the E-field associated with the object has a magnitude of 400 v/m at its surface. A small test charge is then imagined to pop into existence inside the metal. The test charge happens to move from the center of the metal sphere to the surface.
a. Where does the excess charge reside?
b. Determine the work done by the E-field during this motion.
Any conductor in electrostatic equilibrium will have all excess charge reside on the surface. In this way the charge can spread out as much as possible. Because of this, there will be no E-field on the interior of the conductor. With no E-field on the interior, there can be no work done by the field during this migration.
2. (moderate) If the object mentioned in question #1 was shaped like a submarine, make a sketch that includes the E-field at the surface in various locations. Be sure to include relative magnitude information by making the E-field vectors have appropriate lengths. Explain the details of the sketch.
The conductor will have E-field vectors perpendicular to the surface at any point. Because the surface has excess negative charge, the vectors will point inward, toward the surface. The strength of the E-field will be greatest at the ends of the submarine shape, where the curvature is the greatest. The charge density is highest at these locations because these electrons will feel less force along the surface compared to electrons located in flatter areas. Of course, they will feel a greater force off the surface in comparison. Higher charge density produces a stronger field, and the vectors will be longer at the ends.
3. (moderate) Explain why it is safer inside a car during a storm rather than in an open field. Explain the theory that supports your answer.
The metal body of the car will produce a shielding effect. Generally, the base of the clouds is negative and the surface of the Earth is positive during storms. The metal car body will polarize in the E-field caused by the storm clouds. The mobile charge in the car body will migrate due to this external E-field, creating its own E-field that will cancel out the external E-field within the metal shell of the car. With no E-field inside the car, the passengers are much safer during the storm. No such protection is found in an open field. The best thing to do if one is in the open is to lie flat on the ground and spread out to maximize your surface area and minimize the charge density on the surface of your body.
4. (moderate) Make a graph that shows electric potential as a function of distance from the center of a charged solid metal sphere (excess q = 3.7x10-9 C and R = 0.05 m). Calculate maximum potential for this system and record that value on the graph. Also, calculate the electric potential difference between a point found at R/2 and a point found at 2R from the center.
The electric potential inside a conductor in equilibrium is the same as on the surface. This is true due to the equipotential nature of the interior (no E-field means no change in potential). The potential at all points from the center to the surface is:
V = kq/R
V = (9x109)(3.7x10-9)/0.05
V = 666 volts
The potential falls on the inverse with distance once one is off the surface. The equation that determines the potential at points off the surface is:
V = kq/r
For r = 2R:
V = (9x109)(3.7x10-9)/(0.1)
V = 333 volts
The potential difference between the points is -333 volts. This is V at 2R relative to V at R/2.
5. (easy) View the video below. Explain why the discharge from the Tesla Coil hits the tip of the grounding rod.
The pointed tip of the grounding rod can accumulate a higher charge density than other parts of the rod due to its geometry. Thus, the charge on the Tesla Coil is most attracted to this point, just like a lightning strike is attracted to a lightning rod.