Prove that a closed equipotential surface with no charge within itself must enclose an equipotential volume.

Vedclass pdf generator app on play store
Vedclass iOS app on app store
(N/A) Consider a closed equipotential surface $S$ that encloses a volume $V$ containing no charge $(q_{in} = 0)$.
Assume, for the sake of contradiction, that the potential inside the volume is not constant. If the potential varies, there must exist a non-zero electric field $\vec{E}$ within the volume, given by the relation $\vec{E} = -\nabla V$.
According to Gauss's Law, the flux of the electric field through any closed surface is proportional to the enclosed charge: $\oint_S \vec{E} \cdot d\vec{A} = \frac{q_{in}}{\epsilon_0}$.
Since $q_{in} = 0$, the net flux through the surface must be zero.
If the potential were to vary inside, the electric field lines would have to originate or terminate on charges within the volume. However, since there is no charge inside, any field line entering the volume must also exit it, or the field must be zero everywhere.
If the potential were not constant, there would be local maxima or minima of potential inside the region. According to the properties of harmonic functions (Laplace's equation $\nabla^2 V = 0$), the potential cannot have a local maximum or minimum in a charge-free region.
Therefore, the potential must be constant throughout the volume, meaning the entire volume is equipotential.

Explore More

Similar Questions

The diagrams below show regions of equipotentials. $A$ positive charge $q$ is moved from $A$ to $B$ in each diagram.

Assertion $(A):$ $A$ spherical equipotential surface is not possible for a point charge.
Reason $(R):$ $A$ spherical equipotential surface is possible inside a spherical capacitor.

Write the characteristics of equipotential surface.

$A$ hollow conducting sphere is placed in an electric field produced by a point charge at point $P$. If the electric potentials at points $A$,$B$,and $C$ are $V_A$,$V_B$,and $V_C$ respectively,as shown in the figure,then:

Describe schematically the equipotential surfaces corresponding to
$(a)$ a constant electric field in the $z$-direction,
$(b)$ a field that uniformly increases in magnitude but remains in a constant (say,$z$) direction,
$(c)$ a single positive charge at the origin,and
$(d)$ a uniform grid consisting of long equally spaced parallel charged wires in a plane.

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial
For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free
For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo