Two hollow spheres (1 and 2) each having charge Q distributed uniformly. A dipole is placed inside a sphere-1.and nothing is placed inside sphere-2. Choose the correct statement regarding electric field `(E)` inside the sphere and electric flux `(phi)` linked with the spheres

Options

A. `phi_1 =0,E_1 =0 and phi_2 =0, E_2 = 0`
B. `phi_1 !=0,E_1 =0 and phi_2 =0, E_2 != 0`
C. `phi_1 !=0,E_1 !=0 and phi_2 !=0, E_2 = 0`
D. `phi_1 !=0,E_1 !=0 and phi_2 !=0, E_2 != 0`

Answer

To solve the problem regarding the electric field (E) inside two hollow spheres and the electric flux (φ) linked with them, we can follow these steps:

### Step 1: Understand the Setup
We have two hollow spheres (Sphere 1 and Sphere 2) with a uniform charge Q distributed on their surfaces. Sphere 1 contains a dipole inside it, while Sphere 2 is empty.

### Step 2: Analyze Electric Flux (φ)
According to Gauss's Law, the electric flux (φ) through a closed surface is given by:
\[ \phi = \frac{Q_{\text{enclosed}}}{\epsilon_0} \]
- **For Sphere 1**: The dipole consists of a positive charge (+q) and a negative charge (-q). When calculating the enclosed charge for the Gaussian surface inside Sphere 1, the total enclosed charge is zero (since +q and -q cancel each other). However, the charge Q on the surface contributes to the electric flux. Thus, the total flux linked with Sphere 1 is:
\[ \phi_1 = \frac{Q}{\epsilon_0} \quad (\text{due to the surface charge}) \]
- **For Sphere 2**: There is no charge inside Sphere 2. The only charge present is the uniform charge Q distributed on its surface. Therefore, the enclosed charge for any Gaussian surface inside Sphere 2 is also zero, leading to:
\[ \phi_2 = \frac{Q}{\epsilon_0} \quad (\text{due to the surface charge}) \]

### Step 3: Analyze Electric Field (E)
Now, we will analyze the electric field inside both spheres:
- **For Sphere 1**: Inside a hollow conductor with a dipole, the electric field is not zero. The dipole creates an electric field, and thus:
\[ E_1 \neq 0 \quad (\text{inside Sphere 1}) \]
- **For Sphere 2**: Since Sphere 2 is empty and has no charge inside it, the electric field inside Sphere 2 is zero:
\[ E_2 = 0 \quad (\text{inside Sphere 2}) \]

### Step 4: Conclusion
From the analysis, we have:
- Electric flux for Sphere 1 (φ1) is non-zero.
- Electric field inside Sphere 1 (E1) is non-zero.
- Electric flux for Sphere 2 (φ2) is non-zero.
- Electric field inside Sphere 2 (E2) is zero.

Thus, the correct statements regarding electric field and electric flux are:
- φ1 ≠ 0, E1 ≠ 0, φ2 ≠ 0, E2 = 0.

### Final Answer
The correct option is: φ1 ≠ 0, E1 ≠ 0, φ2 ≠ 0, E2 = 0.

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