Question

The electric field $E$ is measured at a point $P(0,0,d)$ generated due to various charge distributions and the dependence of $E$ on $d$ is found to be different for different charge distributions.
Column-II describes different electric charge distributions, along with their locations. Match the functions in column-I with the related charge distributions in column-II.

column -I	column-II
(A) $E$ is independent of $d$	(P) A point charge $Q$ at the origin
(B) $E\propto \frac{1}{d}$	(Q) A small dipole with point charges $Q$ at $(0,0,l)$ and $-Q$ at $(0,0,-l)$. Take $2l<<<d$.
(C) $E\propto \frac{1}{d^2}$	(R) An infinite line charge coincident with the x- axis, with uniform linear charge density $\lambda$
(D) $E\propto \frac{1}{d^3}$	(S) Two infinite wire carrying uniform linear charge density parallel to the x-axis. The one along $(y=0,z=l)$ has a charge density $\lambda$ and the another along $(y=0,z=-l)$ has a charge density $-\lambda$. Take $2l<<d$.
	(T) Infinite plane charge coincident with the xy-plane with uniform surface charge density.

Which of the following option has the correct combiantion considering coulumn-I and column-II.

• A. $A\to S$
• B. $B\to Q,R$
• C. $C\to P,S$
• D. $D\to P$

Answer 1

The correct option is C $C\to P,S$

(P) For a point charge, the electric field is given as,
$E=\frac{kQ}{d^2}$ $\Rightarrow E\propto \frac{1}{d^2}$
$P\to C$

(Q) For a dipole, electric field is given as, $E=\frac{2KP}{d^3}$
$\Rightarrow E\propto \frac{1}{d^3}$
$Q\to D$

(R) For a line charge, the electric field is given as, $E=\frac{2k\lambda }{d}$
$\Rightarrow E\propto \frac{1}{d}$
$R\to B$

(S) Net electric field due to two infinite wires is given as,
$E=\frac{2k\lambda }{d-l}-\frac{2k\lambda }{d+l}$
$E=\frac{4k\lambda l}{d^2[1-\frac{l^2}{d^2}]}$
Since $2l<<d$, $\frac{l^2}{d^2}\approx 0$
$E\propto \frac{1}{d^2}$
$S\to C$

(T) The electric field due to infinite sheet is given as, $E=\frac{\sigma }{2{ϵ}_0}$
Hence $E$ is independent of $d$.
$T\to A$