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 $2 l < < 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