On a thin rod of length L lying on x axis with one end at the origin , as shown in the figure, there is distributed a charge per unit length given by , where k is a constant. (a) Taking the electrostatic potential at infinity to be zero, we have to find V at point P on the y axis. (b) We have to determine the vertical component, , of the electric field at P from the result of part (a) and also by direct calculation. (c) We have to answer why we cannot find , the horizontal component of the electric field at P using the result of part (a). (d) We have to find the distance from the rod along the y axis where the potential is equal to one-half of its value at the left end of the rod.

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(a)

Potential at P due to the charge on the rod can be calculated by integrating contribution from infinitesimal element of length ,

. This definition of the potential ensures that the potential at is zero.

.

For performing the integration, we make the substitution

Therefore,

The y-component of the electric field at P can be obtained by differentiating with respect to y. that is

(b)

We will next calculate directly

For performing the integration, we once again make the substitution

We get

(c)

Potential at the left end of the rod will be

We next find y for which

Or