Question

Diameter of two spheres of metal are 6 cm and 4 cm. They are charged to the same potential. Find out the ratio of the surface densities of charge on the sphere.

Answer 1

Step 1: Formula for Surface Charge Density

The surface charge density \( \sigma \) on a sphere is given by:

\[ \sigma = \frac{Q}{A} \] where \( Q \) is the charge and \( A \) is the surface area of the sphere.

For a sphere, the surface area is: \[ A = 4\pi r^2 \] Since both spheres are charged to the same potential, the charge is proportional to the radius.

Step 2: Ratio of Surface Charge Densities

The ratio of surface charge densities for the two spheres is: \[ \frac{\sigma_1}{\sigma_2} = \frac{Q_1 / A_1}{Q_2 / A_2} = \frac{r_2^2}{r_1^2} \]

Given the diameters are 6 cm and 4 cm, the radii are 3 cm and 2 cm respectively.

\[ \frac{\sigma_1}{\sigma_2} = \frac{3^2}{2^2} = \frac{9}{4} \]

Therefore, the ratio of surface charge densities is: \[ \sigma_1 : \sigma_2 = 9 : 4 \]