Five positive charges each having charge q are placed at the vertices of a pentagon as shown in the figure. The electric potential (V) and the electric field $$(\overrightarrow{E})$$ at the center O of the pentagon due to these five positive charges are:

Each charge is at same distance

r

from center.

Potential is scalar, so potentials add directly.

Potential due to one charge:

$$V_1=\frac{kq}{r}$$

Due to five charges:

$$V=5\frac{kq}{r}$$

Now electric field:

Each charge produces field at center of magnitude

$$E_1=\frac{kq}{r^2}$$

Directions of these five fields are symmetrically spaced by

$$72^{\circ}$$

Because of symmetry, vector sum of all fields is zero.

so

$$E=0$$