Prove that a closed equipotential surface with no charge within itself must enclose an equipotential volume.
Prove that a closed equipotential surface with no charge within itself must enclose an equipotential volume.
Let us assume that in a closed equipotential surface with no charge the potential is changing from position to position. Let the potential just inside the surface is different to that of the surface causing in a potential gradient $\left(\frac{d \mathrm{~V}}{d r}\right)$. It means $\mathrm{E} \neq 0$ electric field comes into existence which is given by as $\mathrm{E}=-\frac{d \mathrm{~V}}{d r}$.
It means, there will be field lines pointing inwards or outwards from the surface. These lines cannot be again on the surface as the surface is equipotential. It is possible only when the other end of the field lines are originated from the charges inside. This contradicts the original assumption. Hence, the entire volume inside must be equipotential.
Similar Questions
What is an equipotential surface ? Draw an equipotential surfaces for a
$(1)$ single point charge
$(2)$ charge $+ \mathrm{q}$ and $- \mathrm{q}$ at few distance (dipole)
$(3)$ two $+ \mathrm{q}$ charges at few distance
$(4)$ uniform electric field.
What is an equipotential surface ? Draw an equipotential surfaces for a
$(1)$ single point charge
$(2)$ charge $+ \mathrm{q}$ and $- \mathrm{q}$ at few distance (dipole)
$(3)$ two $+ \mathrm{q}$ charges at few distance
$(4)$ uniform electric field.
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
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
Three infinitely long linear charges of charge density $\lambda $ , $\lambda $ and $-2\lambda $ are placed in space. A point in space is specified by its perpendicular distance $r_1 , r_2 $ and $ r_3$ respectively from the linear charges. For the points which are equipotential
Three infinitely long linear charges of charge density $\lambda $ , $\lambda $ and $-2\lambda $ are placed in space. A point in space is specified by its perpendicular distance $r_1 , r_2 $ and $ r_3$ respectively from the linear charges. For the points which are equipotential
Equipotential surfaces associated with an electric field which is increasing in magnitude along the $x$-direction are
Equipotential surfaces associated with an electric field which is increasing in magnitude along the $x$-direction are
- [AIIMS 2004]
Two conducting hollow sphere of radius $R$ and $3R$ are found to have $Q$ charge on outer surface when both are connected with a long wire and $q'$ charge is kept at the centre of bigger sphere. Then which one is true
Two conducting hollow sphere of radius $R$ and $3R$ are found to have $Q$ charge on outer surface when both are connected with a long wire and $q'$ charge is kept at the centre of bigger sphere. Then which one is true