Problem 53 - A Complete Resource Book in Physics for JEE Main

Question

Two conducting spheres of radii \(r_{1}\) and \(r_{2}\) are at the same potential. The ratio of their charges is (A) \(\left(\frac{r_{1}^{2}}{r_{2}^{2}}\right)\) (B) \(\left(\frac{r_{2}^{2}}{r_{1}^{2}}\right)\) (C) \(\frac{r_{1}}{r_{2}}\) (D) \(\frac{r_{2}}{r_{1}}\)

Step-by-Step Solution

Answer

The ratio of charges of the two conducting spheres at the same potential is (C) \(\frac{r_1}{r_2}\).

Step 1: Write down the formula for the potential of a conductor

The potential V of a conducting sphere with charge Q and radius r is given by the formula: \[ V = \frac{kQ}{r} \] where k is the electrostatic constant.

Step 2: Set the potentials equal

Since both spheres are at the same potential, we can set their potentials equal to each other, which gives us the equation: \[ \frac{kQ_1}{r_1} = \frac{kQ_2}{r_2} \]

Step 3: Solve for the ratio of charges

Now, we need to solve the above equation for the ratio of charges, \(\frac{Q_1}{Q_2}\). To do this, we first cancel out k from both sides: \[ \frac{Q_1}{r_1} = \frac{Q_2}{r_2} \] Now, we can rearrange this equation to get the ratio of charges: \[ \frac{Q_1}{Q_2} = \frac{r_1}{r_2} \] So the correct answer is (C) \(\frac{r_1}{r_2}\).