A uniform electric field in positive `x`-direction on exists in space. An uncharged solid conducting sphere is now placed in region of this uniform electric field. The centre of this sphere of radius `R` lies at origin. Then electric potential at any point on `y-z` plane (i.e. `x=0` plane) due to only induced charged on surface of sphere.

Options

A. A. will be positive
B. B. will be negative
C. C. will be zero
D. D. may be positive, zero or negative

Explanation

To find the electric potential at any point on the y-z plane (where \( x = 0 \)) due to the induced charges on the surface of an uncharged solid conducting sphere placed in a uniform electric field, we can follow these steps:

### Step 1: Understand the Setup
We have a uniform electric field directed along the positive x-direction. When an uncharged conducting sphere is placed in this field, charges within the sphere will redistribute themselves due to the influence of the electric field.

**Hint:** Visualize the electric field lines and how they interact with the conducting sphere.

### Step 2: Charge Redistribution
Due to the uniform electric field, positive charges will accumulate on the side of the sphere facing the direction of the electric field (the right side, in this case), while negative charges will accumulate on the opposite side (the left side). This creates a dipole-like charge distribution on the sphere.

**Hint:** Remember that conductors allow free movement of charges, which leads to this redistribution.

### Step 3: Analyze the Electric Potential
The electric potential \( V \) at a point in space due to a charge distribution is given by the formula:
\[ V = k \int \frac{dq}{r} \]
where \( k \) is Coulomb's constant, \( dq \) is the charge element, and \( r \) is the distance from the charge element to the point where the potential is being calculated.

**Hint:** Consider the symmetry of the problem when calculating the potential.

### Step 4: Consider the Y-Z Plane
In the y-z plane (where \( x = 0 \)), the induced charges on the sphere will create an electric potential. However, due to the symmetry of the charge distribution (equal positive and negative charges on opposite sides), the contributions to the potential from these charges will cancel out at any point in the y-z plane.

**Hint:** Think about how the potential due to a dipole behaves in relation to its axis.

### Step 5: Conclusion
Since the potential due to the induced charges on the surface of the sphere cancels out at every point in the y-z plane, the electric potential at any point on the y-z plane is zero.

\[ V = 0 \quad \text{for all points in the y-z plane} \]

**Final Answer:** The electric potential at any point on the y-z plane due to the induced charges on the surface of the sphere is zero.