A conducting sphere of radius `R_(2)` has a spherical cavity of radius `R_(1)` which is non concentric with the sphere. A point charge `q_(1)` is placed at a distance r from the centre of the cavity. For this arrangement, answer the following questions.
Electric field (magnitude only) at the center of the cavity will be :

Options

A. zero
B. `(Kq_(1))/(r_(2))`
C. `(Kq_(1))/(r^(2))-(Kq_(1))/(R_(1)^(2))`
D. None of these

Explanation

To solve the problem, we need to analyze the electric field at the center of the cavity of a conducting sphere with a point charge placed inside it. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Setup
We have a conducting sphere of radius \( R_2 \) with a spherical cavity of radius \( R_1 \). A point charge \( q_1 \) is placed at a distance \( r \) from the center of the cavity. Since the cavity is non-concentric with the sphere, the charge is not at the center of the cavity.

**Hint:** Visualize the arrangement to understand the positions of the charge and the cavity.

### Step 2: Induced Charge on the Cavity
When the point charge \( q_1 \) is placed inside the cavity, it will induce a charge on the inner surface of the cavity. The induced charge will be equal in magnitude and opposite in sign to the point charge \( q_1 \). Therefore, the inner surface of the cavity will have a charge of \( -q_1 \).

**Hint:** Remember that conductors redistribute their charge in response to internal charges to maintain electrostatic equilibrium.

### Step 3: Charge on the Outer Surface
Since the conducting sphere is neutral initially, the total charge on the outer surface of the sphere will be equal to the charge induced on the inner surface. Thus, the outer surface will have a charge of \( +q_1 \).

**Hint:** The total charge on a conductor must remain constant, and any induced charge must be balanced by an equal and opposite charge elsewhere.

### Step 4: Electric Field Inside the Cavity
Inside a conductor, the electric field is zero. However, within the cavity, the electric field is influenced by the point charge \( q_1 \) and the induced charge \( -q_1 \).

At the center of the cavity, due to symmetry, the contributions from the induced charges on the cavity surface will cancel out. Therefore, the electric field at the center of the cavity will be solely due to the point charge \( q_1 \).

**Hint:** Use symmetry arguments to deduce that the net electric field contributions from the induced charges will cancel out.

### Step 5: Calculate the Electric Field
The electric field \( E \) at a distance \( r \) from a point charge \( q_1 \) is given by:
\[ E = \frac{k |q_1|}{r^2} \]
However, since we are interested in the electric field at the center of the cavity, we need to consider the distance from the point charge to the center of the cavity. The electric field at the center of the cavity will not be zero but will depend on the distance from the charge to the center.

**Hint:** Remember that the electric field is a vector quantity and depends on the distance from the charge.

### Conclusion
The electric field at the center of the cavity is not zero and is determined by the distance to the point charge \( q_1 \). The electric field will be directed away from the charge if \( q_1 \) is positive and towards the charge if \( q_1 \) is negative.

### Final Answer
The electric field (magnitude only) at the center of the cavity will be given by:
\[ E = \frac{k |q_1|}{(r')^2} \]
where \( r' \) is the distance from the point charge \( q_1 \) to the center of the cavity.