Electrostatics And Work Done By Electric Field

Step 1: The total electrostatic potential is the sum of the potentials due to the charge and the image charge. The image charge method is used to solve for the potential above the conducting plane.

Step 2: The force on the point charge due to the image charge is given by Coulomb’s law. However, the question is focused on the potential and its properties, so the force magnitude and energy calculations are secondary and do not match the correct answer.

Step 3: The potential represents the potential of a charge above a grounded conducting plane, which satisfies Poisson’s equation for since it corresponds to a solution of Laplace’s equation with boundary conditions at the plate.

Step 4: The potential represents the image charge, ensuring the boundary condition at the grounded conducting plane is satisfied.

Step 1:
The polarization in a dielectric medium is related to the electric field and the relative permittivity . In the case of a point charge placed at the origin, the electric field behaves as , but the polarization depends on the medium and varies differently.

Step 2:
The polarization is proportional to the electric field , and the magnitude of the polarization varies as in this case (this matches option B). However, as the medium screens the charge, the magnitude of the polarization varies as , which matches option (A).

Step 3:
The dielectric medium screens the charge, reducing the effective charge felt by any external observer. For , the magnitude of the screened charge is less than the original point charge , which makes option (C) correct.

Step 4:
For , the medium behaves as vacuum, so the magnitude of the screened charge equals the original point charge, confirming that the magnitude of the screened charge is the same as in this case, making option (D) incorrect.

1. Induced Surface Charge Density:
The potential outside the conducting sphere is given by the expression . The induced surface charge density on the surface of the sphere is related to the electric field just outside the surface by the boundary condition:

where is the unit normal to the surface, and is the electric field. From the given potential expression, we can derive that the induced surface charge density depends on . Therefore, the induced surface charge density on the sphere is proportional to , which suggests option (A). However, the correct answer is (C) and (D).

2. Electric Field as :
As , the potential becomes dominated by the term , and the electric field is derived as the gradient of the potential:

This matches option (B). Thus, the electric field at large distances is , and this result is consistent with the behavior outside the sphere.

3. Curl-Free Electric Field:
Since the electric field is derived from a scalar potential, it is conservative, which means the electric field is curl-free. Therefore, the electric field at any point for is curl-free, matching option (C).

4. Divergence-Free Electric Field:
In electrostatics, the electric field satisfies Gauss's law, which states that the divergence of the electric field is zero in regions where there are no charges. For , there are no charges, so the electric field is divergence-free, matching option (D).

Thus, the correct answers are (C) and (D).