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Chapter 13: Problem 53

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) \(\frac{r_{2}}{r_{1}}\)

Short Answer

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

Step by step solution

01

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.
02

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: \[ \frac{kQ_1}{r_1} = \frac{kQ_2}{r_2} \]
03

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) \(\frac{r_1}{r_2}\).

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