Question

A dipole is completely enclosed by a spherical surface. Describe how the total electric flux through this surface varies with the strength of the dipole.

Short answer

Answer: The total electric flux through a spherical surface enclosing a dipole does not depend on the strength of the dipole, as it is always zero regardless of the strength of the dipole.

Step-by-step solution

Definition of Electric Flux

Electric flux is a measure of the electric field's effect over an area. Mathematically, it is defined as the integral of the electric field (E) over a closed surface (S), which can be written as: \[ \Phi_E = \oint_S \vec{E} \cdot d\vec{A} \] where \(\Phi_E\) is the electric flux, \(\vec{E}\) is the electric field vector, and \(d\vec{A}\) is the infinitesimal area vector lying on the surface.

Apply Gauss's Law for a Dipole

Gauss's Law states that the total electric flux through a closed surface is equal to the charge enclosed within the surface divided by the vacuum permittivity \(\epsilon_0\): \[ \oint_S \vec{E} \cdot d\vec{A} = \frac{Q_\text{enclosed}}{\epsilon_0} \] In the case of a dipole, the system consists of two equal and opposite charges (+Q and -Q) separated by some distance (d). Since the positive and negative charges cancel each other out (Q - Q = 0), the net enclosed charge within the spherical surface is zero.

Total Electric Flux for a Dipole Enclosed by Spherical Surface

Given that the net enclosed charge is zero, we can now use Gauss's Law to calculate the total electric flux through the spherical surface. Since there is no net enclosed charge, the total electric flux through the surface will be zero: \[ \oint_S \vec{E} \cdot d\vec{A} = \frac{Q_\text{enclosed}}{\epsilon_0} = \frac{0}{\epsilon_0} = 0 \] Therefore, the total electric flux through the spherical surface enclosing a dipole is zero.

Dependence of Electric Flux on Dipole Strength

When analyzing the dependence of electric flux on the strength of the dipole, it is important to note that a stronger dipole is achieved by either increasing the charge on each end or the distance between the charges. However, as we derived in the previous steps, the total electric flux through a spherical surface enclosing a dipole is always zero, regardless of the strength of the dipole. So, the total electric flux through the spherical surface does not depend on the strength of the dipole.