A point charge is kept at the centre of a metallic insulated spherical shell. Then

Options

A. Electric field out side the sphere is zero
B. Electric field inside the sphere is zero
C. Net induced charge on the sphere is zero
D. Electric potential inside the sphere is zero.

Answer

To solve the problem of a point charge placed at the center of a metallic insulated spherical shell, we can follow these steps:

### Step 1: Understand the Configuration
We have a point charge \( q \) located at the center of a metallic insulated spherical shell. The shell is conductive and insulated, meaning it can redistribute charges but does not allow external electric fields to influence the charge distribution.

**Hint:** Remember that in conductors, charges can move freely and will redistribute themselves in response to an electric field.

### Step 2: Induced Charges on the Inner Surface
Due to the presence of the point charge \( q \) at the center, the inner surface of the spherical shell will have an induced charge. This induced charge will be equal in magnitude but opposite in sign to the point charge, resulting in an induced charge of \( -q \) on the inner surface.

**Hint:** The induced charge on the inner surface of a conductor is always equal and opposite to the charge inside it.

### Step 3: Induced Charges on the Outer Surface
Since the shell is neutral overall and has an induced charge of \( -q \) on the inner surface, the outer surface of the shell must have an induced charge of \( +q \) to maintain the neutrality of the entire shell.

**Hint:** The total charge of a conductor must remain constant; any induced charge on one surface must be balanced by an equal and opposite charge on another surface.

### Step 4: Electric Field Inside the Shell
Inside the metallic shell (between the inner surface and the outer surface), the electric field is zero. This is because the charges on the inner surface redistribute themselves in such a way that they cancel out any electric field within the conducting material.

**Hint:** In electrostatics, the electric field inside a conductor in electrostatic equilibrium is always zero.

### Step 5: Electric Field Outside the Shell
To find the electric field outside the shell, we can use Gauss's law. The total charge enclosed by a Gaussian surface outside the shell is \( q \) (the point charge) plus \( -q \) (the induced charge on the inner surface) plus \( +q \) (the induced charge on the outer surface). This results in a net charge of \( q \) enclosed by the Gaussian surface.

**Hint:** Use Gauss's law, which states that the electric flux through a closed surface is proportional to the enclosed charge.

### Step 6: Conclusion
The electric field outside the shell is not zero and behaves as if there is a point charge \( q \) located at the center of the shell. The net induced charge on the shell itself is zero, as the positive and negative charges balance each other out.

**Final Answer:** The net induced charge on the spherical shell is zero.