Problem 106 - A Complete Resource Book in Physics for JEE Main

Question

Four charges, all of the same magnitude, are placed at the four corners of a square. At the centre of the square, the potential is \(V\) and the field is \(E\). By suitable choices of the signs of the four charges, which of the following can be obtained? (A) \(V=0, E=0\) (B) \(V=0, E \neq 0\) (C) \(V \neq 0, E=0\) (D) \(V \neq 0, E \neq 0\)

Step-by-Step Solution

Answer

All options A, B, C and D are possible with suitable choices of the signs of the four charges.

Step 1 - Consider the Case for \(V=0, E=0\)

To achieve \(V=0\) and \(E=0\), the charges at the ends of each diagonal have to be identical (either both positive or both negative), ensuring equal and opposite electric fields at the center, which then cancel out each other. Similarly, the electric potential at the center of the square would be zero, since equal and opposite potentials cancel out. Hence option (A) is possible.

Step 2 - Consider the Case for \(V=0, E \neq 0\)

If we want the electric field not to be zero, but the potential to be zero, then two adjacent charges should be of the same kind and the other two should be of the opposite kind (e.g., negative, negative, positive, positive). This results in a non-zero electric field but a net electric potential of zero at the center of the square. Hence, option (B) is also possible.

Step 3 - Consider the Case for \(V \neq 0, E=0\)

In order for the electric potential to be non-zero while the electric field is zero, all four charges should be the same. This would result in a zero electric field due to cancellation but a non-zero potential due to the accumulation of effects from the like charges. Therefore, option (C) is possible.

Step 4 - Consider the Case for \(V \neq 0, E \neq 0\)

To have both a non-zero electric field and a non-zero electric potential, three charges should be the same and one different (eg. positive, positive, positive, negative). This would provide a non-zero net electric field and potential. Hence, option (D) is possible.