Electrostatics — JEE Main · 2012 · 29 Apr 2012 · Offline Question with Solution

This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statements.
An insulating solid sphere of radius

R

has a uniformly positive charge density

ρ

. As a result of this uniform charge distribution there is a finite value of electric potential at the centre of the sphere, at the surface of the sphere and also at a point out side the sphere. The electric potential at infinity is zero.
Statement 1: When a charge q is taken from the centre to the surface of the sphere, its potential energy changes by

\frac{qp}{3 ε _{0}}

Statement 2 : The electric field at a distance

r (r < R)

from the centre of the sphere is

\frac{ρ r}{3 ε _{0}}

(A)Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for statement 1.
(B)Statement 1 is true, Statement 2 is false
(C)Statement 1 is false, Statement 2 is true✓ Correct
(D)Statement 1 is true, Statement 2 is the correct explanation for statement 1

Answer: (C)

Solution

Gauss's law at a radius

r < R

inside the uniformly charged sphere gives

E \cdot 4 π r^{2} = \frac{ρ \cdot \frac{4}{3} π r ^{3}}{ε _{0}}

, so

E = \frac{ρ r}{3 ε _{0}}

; the second statement is correct. The potential falls only slightly from centre to surface, and the change in potential energy of a charge

q

carried between them works out to have magnitude

\frac{q ρ R ^{2}}{6 ε _{0}}

, carrying the dimensions of charge times density times length squared. The expression

\frac{q ρ}{3 ε _{0}}

in the first statement is missing the

R^{2}

factor and so cannot be right; the first statement is false. The correct description is that the first statement is false while the second is true.

JEE Main · 2012 · 29 Apr 2012 · Offline · Single correct option

JEE Main · 2012 · 29 Apr 2012 · Offline · Electrostatics

Solution

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