An electric dipole is placed at the centre of a sphere. Select the correct alternative(s).

Options

A. The flux of electric field through the sphere is zero
B. The electric field is zero at every point of the sphere
C. The electric field is not zero anywhere on the sphere
D. The electric field is zero on a circle on the sphere

Answer

To solve the problem regarding the electric dipole placed at the center of a sphere, we can follow these steps:

### Step 1: Understand the Electric Dipole
An electric dipole consists of two equal and opposite charges separated by a distance. For example, we can represent it as +q and -q.

### Step 2: Apply Gauss's Law
According to Gauss's Law, the electric flux (Φ) through a closed surface is given by the equation:
\[ \Phi = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where \(Q_{\text{enc}}\) is the total charge enclosed by the surface and \(\epsilon_0\) is the permittivity of free space.

### Step 3: Determine the Charge Enclosed
In our case, the dipole consists of two charges (+q and -q). When we consider a sphere centered at the dipole, the total charge enclosed by the sphere is:
\[ Q_{\text{enc}} = +q + (-q) = 0 \]

### Step 4: Calculate the Electric Flux
Since the total charge enclosed by the sphere is zero, we can substitute this into Gauss's Law:
\[ \Phi = \frac{0}{\epsilon_0} = 0 \]
Thus, the electric flux through the sphere is zero.

### Step 5: Analyze the Electric Field Inside the Sphere
The electric field (E) due to an electric dipole at a point in space can be expressed as:
\[ E = \frac{K \cdot p}{r^3} \cdot (3 \cos^2 \theta - 1) \]
where \(K\) is a constant, \(p\) is the dipole moment, \(r\) is the distance from the dipole, and \(\theta\) is the angle between the dipole axis and the line connecting the dipole to the point of interest.

### Step 6: Evaluate the Electric Field at Points on the Sphere
For any point on the surface of the sphere, the electric field will not be zero. The expression for the electric field indicates that it depends on the angle \(\theta\), and since \(\cos^2 \theta\) is always greater than or equal to zero, the electric field will always have a non-zero value at any point on the sphere.

### Conclusion
From the analysis, we can conclude:
1. The electric flux through the sphere is zero (due to the enclosed charge being zero).
2. The electric field is not zero at any point on the surface of the sphere.

### Correct Alternatives
Thus, the correct alternatives based on the analysis are:
- The flux of electric field through the sphere is zero.
- The electric field is not zero at every point of the sphere.