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 de

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. [2018]

	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 $λ$
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 $+ λ$ and the one along $(y = 0, z = - l)$ has a charge density $- λ$ . Take $2 l ≪ d$
		5.	Infinite plane charge coincident with the $x y$ -plane with uniform surface charge density

Smart Achievers · Accepted Answer

(2)

$For a point charge$ $E = \frac{k Q}{d^{2}}$ i.e., $E \propto \frac{1}{d^{2}}$

$and for a dipole$ $E = \frac{k p}{d^{3}}$ i.e., $E \propto \frac{1}{d^{3}}$

$For an infinite long line charge$ $E = \frac{2 k λ}{d}$ i.e., $E \propto \frac{1}{d}$

$For two infinite wires carrying uniform linear charge density$

$E = \frac{2 k λ}{r} \cos α = \frac{2 k λ}{\sqrt{d^{2} + ℓ^{2}}} \times \frac{ℓ}{\sqrt{d^{2} + ℓ^{2}}} = \frac{2 k λ ℓ}{d^{2} + ℓ^{2}}$

or, $E \propto \frac{1}{d^{2}}$ $∵$ $2 ℓ ≪ d$

$For infinite plane charge$ $E = \frac{σ}{2 ε_{0}}$ i.e., $E is independent of d$

Solution

1 P → 5; Q → 3, 4; R → 1; S → 2

2 P → 5; Q → 3; R → 1, 4; S → 2

3 P → 5; Q → 3; R → 1, 2; S → 4

4 P → 4; Q → 2, 3; R → 1; S → 5

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