Question

A broad metal plate is connected to earth through a conducting wire as shown. An electron flies with constant velocity along a straight line above the plate at a distance much less than the linear dimensions of the plate. If the current I flowing from the earth to the plate is considered to be positive, then which of the following graph correctly depicts the variation of current I with time?

• A.
• B.
• C.
• D.

Answer 1

Answer: (B)

B

Answer 2

The problem describes an electron flying with constant velocity along a straight line above a broad metal plate connected to earth. We need to determine the variation of current I flowing from the earth to the plate with time.

1. Induced Charge: When an electron (a negative charge, -e) moves near a grounded conducting plate, it induces positive charges on the plate. This phenomenon can be understood using the method of images. For a point charge q above an infinite grounded conducting plane, the electric field in the region above the plane is equivalent to that produced by the original charge q and an image charge -q located symmetrically below the plane. Since our electron has charge q = -e, its image charge will be -(-e) = +e.

2. Total Induced Charge on the Plate: The total induced charge on an infinite grounded conducting plane due to a point charge q is -q. Therefore, for an electron with charge -e, the total induced charge Q_ind on the plate will be -(-e) = +e. This means that as the electron passes, a net positive charge is induced on the plate.

3. Variation of Induced Charge with Time:

   • Electron approaching (t < 0): When the electron is far away from the plate (e.g., far to the left), its influence is negligible, so the induced charge Q_ind on the plate is approximately zero. As the electron approaches the plate, positive charges are attracted from the earth onto the plate. Thus, Q_ind increases from zero.
   • Electron directly above (t = 0): When the electron is directly above the plate, the attractive force is strongest, and the induced positive charge Q_ind reaches its maximum value, which is +e. At this point, the rate of change of induced charge is momentarily zero.
   • Electron moving away (t > 0): As the electron moves away from the plate (e.g., to the right), its influence diminishes. The positive charges on the plate are no longer strongly attracted and flow back to the earth. Thus, Q_ind decreases from its maximum value back to zero.

   A graph of Q_ind versus time t would look like a bell-shaped curve, symmetric about t=0, rising from 0 to +e and then falling back to 0.

4. Current I: The current I flowing from the earth to the plate is defined as the rate of change of charge on the plate: I = dQ_ind / dt.

   • Electron approaching (t < 0): Q_ind is increasing, so dQ_ind/dt > 0. This means the current I is positive.
   • Electron directly above (t = 0): Q_ind is at its maximum, so dQ_ind/dt = 0. This means the current I is zero.
   • Electron moving away (t > 0): Q_ind is decreasing, so dQ_ind/dt < 0. This means the current I is negative.

   Therefore, the current I starts at zero, increases to a positive maximum, passes through zero when the electron is directly overhead, then decreases to a negative minimum, and finally returns to zero as the electron moves far away. This shape is characteristic of the derivative of a bell-shaped curve.

5. Matching with Graphs:

   • Graph (A) shows a negative current pulse followed by a positive current pulse.
   • Graph (B) shows a positive current pulse followed by a negative current pulse. This matches our analysis.
   • Graph (C) shows a more complex pattern.
   • Graph (D) shows a constant current, then zero.

Thus, graph (B) correctly depicts the variation of current I with time.