Calculate potential on the axis of a disc of radius $R$ due to a charge $Q$ uniformly distributed on its surface.
Calculate potential on the axis of a disc of radius $R$ due to a charge $Q$ uniformly distributed on its surface.
Let us consider a point $\mathrm{P}$ on the axis of the disc at a distance $x$ from the centre of the disc. Let the disc is divided into a numerous charged rings as shown in figure.
Let radius of ring $r$, width $d r$ and charges $d q$,
$\therefore \sigma d \mathrm{~A}=\sigma 2 \pi r d r$$..(1)$
Potential at $\mathrm{P}$,
$d \mathrm{~V}=\frac{k d q}{r}$
Charge on the ring $d q=+\sigma\left[\pi(r+d r)^{2}-\pi r^{2}\right]$
$\therefore d q=+\sigma \pi\left[(r+d r)^{2}-r^{2}\right]$
$\quad=+\sigma \pi\left[r^{2}+2 r d r+d r^{2}-r^{2}\right]$
$\quad=+\sigma \pi\left[2 r d r+d r^{2}\right]$
Neglecting $d r^{2}$ as $d r$ is very small,
$d q=2 \pi r \sigma d r$
and $d \mathrm{~V}=\frac{k d q}{\sqrt{r^{2}+x^{2}}}$$....(2)$
Similar Questions
In a certain charge distribution, all points having zero potential can be joined by a circle $S$. Points inside $S$ have positive potential and points outside $S$ have negative potential. A positive charge, which is free to move, is placed inside $S$
In a certain charge distribution, all points having zero potential can be joined by a circle $S$. Points inside $S$ have positive potential and points outside $S$ have negative potential. A positive charge, which is free to move, is placed inside $S$
A solid conducting sphere having a charge $Q$ is surrounded by an uncharged concentric conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be $V$. If the shell is now given a charge of $-3Q$, the new potential difference between the same two surfaces is......$V$
A solid conducting sphere having a charge $Q$ is surrounded by an uncharged concentric conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be $V$. If the shell is now given a charge of $-3Q$, the new potential difference between the same two surfaces is......$V$
- [IIT 1989]
Four electric charges $+q,+q, -q$ and $-q$ are placed at the comers of a square of side $2L$ (see figure). The electric potential at point $A,$ midway between the two charges $+q$ and $+q,$ is
Four electric charges $+q,+q, -q$ and $-q$ are placed at the comers of a square of side $2L$ (see figure). The electric potential at point $A,$ midway between the two charges $+q$ and $+q,$ is
- [AIPMT 2011]
Twenty seven drops of water of the same size are equally and similarly charged. They are then united to form a bigger drop. By what factor will the electrical potential changes.........$times$
Twenty seven drops of water of the same size are equally and similarly charged. They are then united to form a bigger drop. By what factor will the electrical potential changes.........$times$
Two identical positive charges are placed on the $y$-axis at $y=-a$ and $y=+a$. The variation of $V$ (electric potential) along $x$-axis is shown by graph
Two identical positive charges are placed on the $y$-axis at $y=-a$ and $y=+a$. The variation of $V$ (electric potential) along $x$-axis is shown by graph