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	An 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

Answer: (B)

(i) $E=\frac{KQ}{d^2}\Rightarrow E\propto \frac{1}{d^2}$
(ii) Dipole
$E=\frac{2kp}{d^3}\sqrt{1+3{\cos }^2\theta }$
$E\propto \frac{1}{d^3}$ for dipole
(iii) For line charge
$E=\frac{2k\lambda }{d}$
$E\propto \frac{1}{d}$
(iv) $E=\frac{2K\lambda }{d-l}-\frac{2K\lambda }{d+l}$
$=2K\lambda \left[\frac{d+l-d+l}{d^2-l^2}\right]$
$E=\frac{2K\lambda \left(2l\right)}{d^2\left[1-\frac{l^2}{d^2}\right]}$
$E\propto \frac{1}{d^2}$
(v) Electric field due to sheet
$\in =\frac{\sigma }{2{\in }_0}$
$\in =v$ is independent of r