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- \rm I contains different relations between E and d . List- \rm II describes different electric charge distributions, along with their locations. Match the functions in List- \rm I with the related charge distributions in List- \rm II . .tg border-collapse:collapse;border-spacing:0;border-color:#999; .tg tdfont-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#999;color:#444;background-color:#F7FDFA; .tg thfont-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#999;color:#fff;background-color:#26ADE4; .tg .tg-yttmfont-weight:bold;font-size:22px;text-align:left .tg .tg-pjk9font-size:22px;text-align:left .tg .tg-cyimfont-size:22px;text-align:left;vertical-align:top LIST - I LIST - II P. E is independent of d 1. A point charge Q at the origin Q. E\, \propto \,1/d 2. A small dipole with point charges Q at ( 0,0,l ) and -Q at ( 0,0, - l ). Take 2l \lt \lt d R. E\, \propto \,1/d^2 3. An infinite line charge coincident with the x-axis, with uniform linear charge density \lambda S. E\, \propto \,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 Take 5. Infinite plane charge coincident with the xy -plane with uniform surface charge density

P \to 5;Q \to 3;R \to 1,4;S \to 2

P \to 5;Q \to 3,4;R \to 1;S \to 2

P \to 4;Q \to 3;R \to 1,2;S \to 4

P \to 4;Q \to 2,3;R \to 1;S \to 5

Electrostatics PYQ — JEE Advanced 2018 · Shift 2 · Q51 with Solution

The electric field

E

is measured at a point

P(0,0,d)

generated due to various charge distributions and the dependence of

E

d

is found to be different for different charge distributions. List-

{\rm I}

contains different relations between

E

and

d

. List-

{\rm II}

describes different electric charge distributions, along with their locations. Match the functions in List-

{\rm I}

with the related charge distributions in List-

{\rm II}

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

View written solutionFree

Correct answer: B

Interpret the charge distributions

We need the electric field at $P(0,0,d)$ and its dependence on $d$ .

List-II contains:

Point charge $Q$ at origin.
Small dipole with charges $+Q$ at $(0,0,l)$ and $-Q$ at $(0,0,-l)$ , with $2l \ll d$ .
Infinite line charge along the $x$ -axis with linear charge density $\lambda$ .
Two infinite line charges parallel to the $x$ -axis, one at $z=l$ with $+\lambda$ and one at $z=-l$ with $-\lambda$ .
Infinite plane sheet in the $xy$ -plane with surface charge density $\sigma$ .

We now find how $E$ varies with $d$ in each case.

Case 1: Point charge at origin

Distance of $P(0,0,d)$ from origin is $d$ .

$E=\frac{1}{4\pi\varepsilon_0}\frac{Q}{d^2}$

So,

$E\propto \frac{1}{d^2}$

Hence,

$1 \to R$

Case 2: Small dipole on the $z$ -axis

Point $P$ lies on the axial line of the dipole. For a short dipole, axial field is

$E = \frac{1}{4\pi\varepsilon_0}\frac{2p}{d^3}, \qquad p=Q(2l)$

Thus,

$E\propto \frac{1}{d^3}$

Hence,

$2 \to S$

Case 3: Infinite line charge along the $x$ -axis

The perpendicular distance of $P(0,0,d)$ from the $x$ -axis is $d$ .

Field due to an infinite line charge is

$E=\frac{\lambda}{2\pi\varepsilon_0 d}$

So,

$E\propto \frac{1}{d}$

Hence,

$3 \to Q$

Case 4: Two infinite line charges at $z=\pm l$

One line at $z=l$ has $+\lambda$ , the other at $z=-l$ has $-\lambda$ .

At $P(0,0,d)$ , distances from the two lines are:

$r_+ = d-l, \qquad r_- = d+l$

Field magnitudes:

$E_+ = \frac{\lambda}{2\pi\varepsilon_0(d-l)}, \qquad E_- = \frac{\lambda}{2\pi\varepsilon_0(d+l)}$

Their directions are opposite along the $z$ -axis, so net field magnitude is

$E = \frac{\lambda}{2\pi\varepsilon_0}\left(\frac{1}{d-l}-\frac{1}{d+l}\right)$

$E = \frac{\lambda}{2\pi\varepsilon_0}\cdot \frac{2l}{d^2-l^2}$

For $d \gg l$ ,

$E \approx \frac{\lambda}{2\pi\varepsilon_0}\cdot \frac{2l}{d^2} \propto \frac{1}{d^2}$

So this behaves like

$E\propto \frac{1}{d^2}$

Hence,

$4 \to R$

Case 5: Infinite plane sheet in the $xy$ -plane

Field due to an infinite plane sheet is independent of distance:

$E=\frac{\sigma}{2\varepsilon_0}$

So,

$E \text{ is independent of } d$

Hence,

$5 \to P$

Now match List-I with List-II

$P$ : $E$ independent of $d$ $\to 5$
$Q$ : $E\propto 1/d$ $\to 3$
$R$ : $E\propto 1/d^2$ $\to 1,4$
$S$ : $E\propto 1/d^3$ $\to 2$

So the correct matching is:

$P\to 5,\quad Q\to 3,\quad R\to 1,4,\quad S\to 2$

This corresponds to Option B.

Comparison with stored answer

Stored correct answer: B

Our derived answer: B

They match.

More from Electrostatics

The electric field

E

is measured at a point

P(0,0,d)

generated due to various charge distributions and the dependence of

E

d

is found to be different for different charge distributions. List-

{\rm I}

contains different relations between

E

and

d

. List-

{\rm II}

describes different electric charge distributions, along with their locations. Match the functions in List-

{\rm I}

with the related charge distributions in List-

{\rm II}

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

View written solutionFree

Correct answer: B

Interpret the charge distributions

We need the electric field at $P(0,0,d)$ and its dependence on $d$ .

List-II contains:

Point charge $Q$ at origin.
Small dipole with charges $+Q$ at $(0,0,l)$ and $-Q$ at $(0,0,-l)$ , with $2l \ll d$ .
Infinite line charge along the $x$ -axis with linear charge density $\lambda$ .
Two infinite line charges parallel to the $x$ -axis, one at $z=l$ with $+\lambda$ and one at $z=-l$ with $-\lambda$ .
Infinite plane sheet in the $xy$ -plane with surface charge density $\sigma$ .

We now find how $E$ varies with $d$ in each case.

Case 1: Point charge at origin

Distance of $P(0,0,d)$ from origin is $d$ .

$E=\frac{1}{4\pi\varepsilon_0}\frac{Q}{d^2}$

So,

$E\propto \frac{1}{d^2}$

Hence,

$1 \to R$

Case 2: Small dipole on the $z$ -axis

Point $P$ lies on the axial line of the dipole. For a short dipole, axial field is

$E = \frac{1}{4\pi\varepsilon_0}\frac{2p}{d^3}, \qquad p=Q(2l)$

Thus,

$E\propto \frac{1}{d^3}$

Hence,

$2 \to S$

Case 3: Infinite line charge along the $x$ -axis

The perpendicular distance of $P(0,0,d)$ from the $x$ -axis is $d$ .

Field due to an infinite line charge is

$E=\frac{\lambda}{2\pi\varepsilon_0 d}$

So,

$E\propto \frac{1}{d}$

Hence,

$3 \to Q$

Case 4: Two infinite line charges at $z=\pm l$

One line at $z=l$ has $+\lambda$ , the other at $z=-l$ has $-\lambda$ .

At $P(0,0,d)$ , distances from the two lines are:

$r_+ = d-l, \qquad r_- = d+l$

Field magnitudes:

$E_+ = \frac{\lambda}{2\pi\varepsilon_0(d-l)}, \qquad E_- = \frac{\lambda}{2\pi\varepsilon_0(d+l)}$

Their directions are opposite along the $z$ -axis, so net field magnitude is

$E = \frac{\lambda}{2\pi\varepsilon_0}\left(\frac{1}{d-l}-\frac{1}{d+l}\right)$

$E = \frac{\lambda}{2\pi\varepsilon_0}\cdot \frac{2l}{d^2-l^2}$

For $d \gg l$ ,

$E \approx \frac{\lambda}{2\pi\varepsilon_0}\cdot \frac{2l}{d^2} \propto \frac{1}{d^2}$

So this behaves like

$E\propto \frac{1}{d^2}$

Hence,

$4 \to R$

Case 5: Infinite plane sheet in the $xy$ -plane

Field due to an infinite plane sheet is independent of distance:

$E=\frac{\sigma}{2\varepsilon_0}$

So,

$E \text{ is independent of } d$

Hence,

$5 \to P$

Now match List-I with List-II

$P$ : $E$ independent of $d$ $\to 5$
$Q$ : $E\propto 1/d$ $\to 3$
$R$ : $E\propto 1/d^2$ $\to 1,4$
$S$ : $E\propto 1/d^3$ $\to 2$

So the correct matching is:

$P\to 5,\quad Q\to 3,\quad R\to 1,4,\quad S\to 2$

This corresponds to Option B.

Comparison with stored answer

Stored correct answer: B

Our derived answer: B

They match.

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