Two isolated charged conducting spheres of radii a and b produce the same electric field near their surface. The ratio of electric potentials on their surfaces is

Options

A. `a/b`
B. `b/a`
C. `a^2/b^2`
D. `b^2/a^2`

Explanation

To solve the problem of finding the ratio of electric potentials on the surfaces of two isolated charged conducting spheres of radii \( a \) and \( b \) that produce the same electric field near their surfaces, we can follow these steps:

### Step 1: Understand the Electric Field Near the Surface
The electric field \( E \) just outside the surface of a charged conducting sphere is given by the formula:
\[ E = \frac{kQ}{r^2} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge on the sphere, and \( r \) is the radius of the sphere.

### Step 2: Set Up the Electric Field Equations
For the two spheres, we can denote:
- Sphere 1 (radius \( a \), charge \( Q_1 \))
- Sphere 2 (radius \( b \), charge \( Q_2 \))

According to the problem, the electric fields at points just outside the surfaces of both spheres are equal:
\[ E_1 = E_2 \]
This gives us the equation:
\[ \frac{kQ_1}{a^2} = \frac{kQ_2}{b^2} \]

### Step 3: Simplify the Equation
Since \( k \) is a constant, we can cancel it out from both sides:
\[ \frac{Q_1}{a^2} = \frac{Q_2}{b^2} \]
Rearranging this gives us:
\[ \frac{Q_1}{Q_2} = \frac{a^2}{b^2} \quad \text{(Equation 1)} \]

### Step 4: Write the Potential Equations
The electric potential \( V \) on the surface of a charged conducting sphere is given by:
\[ V = \frac{kQ}{r} \]
Thus, the potentials for both spheres are:
- For Sphere 1:
\[ V_1 = \frac{kQ_1}{a} \]
- For Sphere 2:
\[ V_2 = \frac{kQ_2}{b} \]

### Step 5: Find the Ratio of Potentials
We want to find the ratio of the potentials on the surfaces of the two spheres:
\[ \frac{V_1}{V_2} = \frac{\frac{kQ_1}{a}}{\frac{kQ_2}{b}} = \frac{Q_1}{Q_2} \cdot \frac{b}{a} \]
Substituting Equation 1 into this expression:
\[ \frac{V_1}{V_2} = \frac{a^2}{b^2} \cdot \frac{b}{a} \]

### Step 6: Simplify the Ratio
Now, simplify the expression:
\[ \frac{V_1}{V_2} = \frac{a^2 \cdot b}{b^2 \cdot a} = \frac{a}{b} \]

### Final Answer
Thus, the ratio of electric potentials on their surfaces is:
\[ \frac{V_1}{V_2} = \frac{a}{b} \]