Two spheres of radius `a` and `b` respectively are charged and joined by a wire. The ratio of electric field of the spheres 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 the electric fields of two charged spheres of radii \( a \) and \( b \) when they are connected by a wire, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Electric Field and Potential:**
The electric field \( E \) due to a charged sphere at its surface is given by the formula:
\[ E = \frac{V}{r} \]
where \( V \) is the electric potential and \( r \) is the radius of the sphere.

2. **Electric Potential of a Charged Sphere:**
The electric potential \( V \) at the surface of a charged sphere with charge \( Q \) is given by:
\[ V = k \frac{Q}{r} \]
where \( k \) is Coulomb's constant.

3. **Electric Field in Terms of Charge and Radius:**
Therefore, the electric field \( E \) at the surface of the sphere can be expressed as:
\[ E = k \frac{Q}{r^2} \]
where \( r \) is the radius of the sphere.

4. **Connecting the Spheres:**
When the two spheres are connected by a wire, they reach the same electric potential. Let’s denote the charges on the spheres as \( Q_a \) for the sphere of radius \( a \) and \( Q_b \) for the sphere of radius \( b \).

5. **Setting the Potentials Equal:**
Since the potentials are equal:
\[ V_a = V_b \]
This gives us:
\[ k \frac{Q_a}{a} = k \frac{Q_b}{b} \]
Simplifying this, we find:
\[ \frac{Q_a}{a} = \frac{Q_b}{b} \]

6. **Finding the Ratio of Electric Fields:**
Now, we can express the electric fields \( E_a \) and \( E_b \) for the two spheres:
\[ E_a = k \frac{Q_a}{a^2} \]
\[ E_b = k \frac{Q_b}{b^2} \]
To find the ratio of the electric fields:
\[ \frac{E_a}{E_b} = \frac{k \frac{Q_a}{a^2}}{k \frac{Q_b}{b^2}} = \frac{Q_a \cdot b^2}{Q_b \cdot a^2} \]

7. **Substituting the Charge Ratio:**
From the equality of potentials, we have:
\[ \frac{Q_a}{Q_b} = \frac{a}{b} \]
Substituting this into the electric field ratio:
\[ \frac{E_a}{E_b} = \frac{(a/b) \cdot b^2}{a^2} = \frac{b}{a} \]

8. **Final Result:**
Thus, the ratio of the electric fields of the two spheres is:
\[ \frac{E_a}{E_b} = \frac{b}{a} \]

### Conclusion:
The ratio of the electric fields of the two spheres is \( \frac{b}{a} \).