New chat PYQ Library Doubts Rank report

Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →

Electrostatics question

2019 · 9 Jan · Shift 1 · Q62

Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.

PYQ Library
JEE Main
Physics
Electrostatics
2019 · 9 Jan · Shift 1 · Q62

Electrostatics question

2019 · 9 Jan · Shift 1 · Q62

JEE MainPhysicsElectrostaticsMCQ+4 / −1

For a uniformly charged ring of radius R, the electric field on its axis has the largest magnitude at a distance h from its center. Then value of h is :

A
${R \over {\sqrt 5 }}$
B
${R \over {\sqrt 2 }}$
C
R
D
R $\sqrt 2$

View written solutionFree

Correct answer: B

Electric field on the axis of a uniformly charged ring

For a ring of radius $R$ carrying total charge $Q$ , the electric field at a point on its axis at distance $x$ from the center is

E(x)=\frac{1}{4\pi\varepsilon_0}\frac{Qx}{(R^2+x^2)^{3/2}}.

We need the value of $x=h$ for which $E$ is maximum.

Maximize the function

Since constants do not affect the maximum, maximize

f(x)=\frac{x}{(R^2+x^2)^{3/2}}.

Differentiate with respect to $x$ :

f'(x)=\frac{d}{dx}\left[x(R^2+x^2)^{-3/2}\right].

Using product rule,

f'(x)=(R^2+x^2)^{-3/2}+x\left(-\frac{3}{2}\right)(R^2+x^2)^{-5/2}(2x).

So,

f'(x)=(R^2+x^2)^{-3/2}-3x^2(R^2+x^2)^{-5/2}.

Take common factor $(R^2+x^2)^{-5/2}$ :

f'(x)=(R^2+x^2)^{-5/2}\big[(R^2+x^2)-3x^2\big].

f'(x)=(R^2+x^2)^{-5/2}(R^2-2x^2).

For maximum, set $f'(x)=0$ :

R^2-2x^2=0

2x^2=R^2

x^2=\frac{R^2}{2}

x=\frac{R}{\sqrt{2}}.

Since distance is positive,

h=\frac{R}{\sqrt{2}}.

Check with options

A: $\dfrac{R}{\sqrt{5}}$ ❌
B: $\dfrac{R}{\sqrt{2}}$ ✅
C: $R$ ❌
D: $R\sqrt{2}$ ❌

Thus the correct option is B.

Comparison with stored answer

Stored correct answer: B

Derived answer: B

They agree.

More from Electrostatics

Three charges + Q, q, + Q are placed respectively, at distance, 0, d/2 and d from the origin, on the x-axis. If the net force experienced by + Q, placed at x = 0, is zero, then value of q is :2019 · MCQ
Charge is distributed within a sphere of radius R with a volume charge density $ρ (r) = \frac{A}{r ^{2}} e^{- \frac{2 r}{s}},$ where A and a are constants. If Q is the total charge of this charge distribution, the…2019 · MCQ
Two point charges q1 $(10 μ C)$ and q2( $-$ 25 $μ$ C) are placed on the x-axis at x = 1 m and x = 4 m respectively. The electric field (in V/m) at a point y = 3 m on y-axis is, [take ${1 \over {4\pi { \in…2019 · MCQ
A uniformly charged ring of radius 3a and total charge q is placed in xy-plane centred at origin. A point charge q is moving towards the ring along the z-axis and has speed u at z = 4a. The minimum value of u such that it crosses the…2019 · MCQ
In free space, a particle A of charge 1 $μ$ C is held fixed at a point P. Another particle B of the same charge and mass 4 $μ$ g is kept at a distance of 1 mm from P. If B is released, then its velocity at a distance of 9 mm from P is…2019 · MCQ
A charge Q is distributed over three concentric spherical shells of radii a, b, c (a < b < c) such that their surface charge densities are equal to one another. The total potential at a point at distance r from their common centre,…2019 · MCQ
Two electric dipoles, A, B with respective dipole moments $d_{A} = - 4 q ai$ and $d_{B} = - 2 q ai$ are placed on the x-axis with a separation R, as shown in the figure. The distance from A at which both of… Includes diagram2019 · MCQ
Charges –q and +q located at A and B, respectively, constitude an electric dipole. Distance AB = 2a, O is the mid point of the dipole and OP is perpendicular to AB. A charge Q is placed at P where OP = y and y >> 2a. The charge Q… Includes diagram2019 · MCQ