Two conducting spheres of radii `r_(1)` and `r_(2)` are equally charged. The ratio of their potentral is-

Options

A. `r_(1)^(2)//r_(2)^(2)`
B. `r_(2)^(2)//r_(1)^(2)`
C. `r_(1)//r_(2)`
D. `r_(2)//r_(1)`

Answer

To solve the problem of finding the ratio of the potentials of two conducting spheres with radii \( r_1 \) and \( r_2 \) that are equally charged, we can follow these steps:

### Step 1: Understand the formula for electric potential
The electric potential \( V \) at the surface of a charged conducting sphere is given by the formula:
\[ V = \frac{kQ}{R} \]
where \( k \) is the Coulomb's constant, \( Q \) is the charge on the sphere, and \( R \) is the radius of the sphere.

### Step 2: Write the expressions for the potentials of both spheres
Let the charge on both spheres be \( Q \). Then, the potentials for the two spheres can be expressed as:
- For the first sphere (radius \( r_1 \)):
\[ V_1 = \frac{kQ}{r_1} \]
- For the second sphere (radius \( r_2 \)):
\[ V_2 = \frac{kQ}{r_2} \]

### Step 3: Find the ratio of the potentials
To find the ratio of the potentials \( V_1 \) and \( V_2 \), we can write:
\[ \frac{V_1}{V_2} = \frac{\frac{kQ}{r_1}}{\frac{kQ}{r_2}} \]
Here, \( kQ \) cancels out:
\[ \frac{V_1}{V_2} = \frac{r_2}{r_1} \]

### Step 4: State the final result
Thus, the ratio of the potentials of the two spheres is:
\[ \frac{V_1}{V_2} = \frac{r_2}{r_1} \]

### Summary
The ratio of the potentials of the two conducting spheres with radii \( r_1 \) and \( r_2 \) that are equally charged is given by:
\[ \frac{V_1}{V_2} = \frac{r_2}{r_1} \]

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