A solid metallic sphere has a charge `+3Q`. Concentric with this sphere is a conducting spherical shell having charge `-Q`. The radius of the sphere is `a` and that of the spherical shell is `b(bgta)` What is the electric field at a distance `R(a lt R lt b)` from the centre

Options

A. `1/(4pi epsi_(0)) Q/r`
B. `1/(4piepsi_(0)) (3Q)/r`
C. `1/(4piepsi_(0)) (3Q)/r^(2)`
D. `1/(4pi epsi_(0)) Q/r^(2)`

Answer

To determine the electric field at a distance \( R \) from the center of a solid metallic sphere with charge \( +3Q \) and a concentric conducting spherical shell with charge \( -Q \), we can use Gauss's law.

### Step-by-Step Solution:

1. **Identify the Region**: We need to find the electric field at a distance \( R \) where \( a < R < b \). This means we are looking at a point that is outside the solid metallic sphere but inside the conducting spherical shell.

2. **Use Gauss's Law**: According to Gauss's law, the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space (\( \epsilon_0 \)):
\[ \Phi_E = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where \( \Phi_E = E \cdot A \) and \( A \) is the surface area of the Gaussian surface.

3. **Choose a Gaussian Surface**: We choose a spherical Gaussian surface of radius \( R \) (where \( a < R < b \)). The area \( A \) of this surface is given by:
\[ A = 4\pi R^2 \]

4. **Determine the Enclosed Charge**: Inside this Gaussian surface, the only charge that contributes to the electric field is the charge of the solid metallic sphere, which is \( +3Q \). The charge on the conducting shell does not contribute to the electric field inside it.

5. **Calculate the Electric Flux**: The electric field \( E \) is uniform over the Gaussian surface, so we can express the electric flux as:
\[ \Phi_E = E \cdot A = E \cdot 4\pi R^2 \]

6. **Set Up the Equation**: From Gauss's law, we have:
\[ E \cdot 4\pi R^2 = \frac{3Q}{\epsilon_0} \]

7. **Solve for the Electric Field \( E \)**:
\[ E = \frac{3Q}{4\pi \epsilon_0 R^2} \]

### Final Result:
The electric field at a distance \( R \) from the center of the sphere, where \( a < R < b \), is given by:
\[ E = \frac{3Q}{4\pi \epsilon_0 R^2} \]