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. List - I contains different relations between E and d. List - II describes different electric charge distributions, along with their locations. Match the functions in List - I with the related charge distributions in List - II

List - I	List - II
(P) E is independent of d	(1) A point charge Q at the origin
(Q) $E\propto \frac{1}{d}$	(2) A small dipole with point charges Q at $(0,0,l)$ and - Q at $(0,0,-l)$. Take $2l<<d$
(R) $E\propto \frac{1}{d^2}$	(3) A infinite line charge coincident with the x-axis, with uniform linear charge density $\lambda$.
(S) $E\propto \frac{1}{d^3}$	(4) Two infinite wires carrying uniform linear charge density parallel to the x-axis. The one along ($y=0,z=l$) has a charge density + $\lambda$ and the one along $(y=0,z=-l)$ has a charge density $-\lambda$. Take $2l<<d$
	(5) Infinite plane charge coincident with the xy-plane with uniform surface charge density

• A. $P\to 5;Q\to 3,4;R\to 1;S\to 2$
• B. $P\to 5;Q\to 3;R\to 1,4;S\to 2$
• C. $P\to 5;Q\to 3;R\to 1,2;S\to 4$
• D. $P\to 4;Q\to 2,3;R\to 1;S\to 5$

Answer 1

The correct option is B $P\to 5;Q\to 3;R\to 1,4;S\to 2$

(1) E = $\frac{KQ}{d^2}\Rightarrow E\propto \frac{1}{d^2}$

(2) Dipole
$E=\frac{2kp}{d^3}\sqrt{1+3{cos}^2\theta }$
$E\propto \frac{1}{d^3}$ for dipole

(3) For line charge
$E=\frac{2k\lambda }{d}$
$⟹E\propto \frac{1}{d}$

(4) E = $\frac{2K\lambda }{d-l}-\frac{2K\lambda }{d+l}$
=$2K\lambda \left[\frac{d+l-d+l}{d^2–l^2}\right]$
$E=\frac{2K\lambda \left(2l\right)}{d^2[1-\frac{l^2}{d^2}]}$
$E\propto \frac{1}{d^2}$

(5) Electric field due to sheet
$E=\frac{\sigma }{2{ϵ}_0}$
$E$ is independent of $d$