This question has statement- 1 and statement- 2. Of the four choices… | JEE Main 2012 (AIEEE)

Q119JEE Main 2012 · AIEEEMedium

Electric Flux & Gauss's Law

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 $\rho$ . As a result of this uniform charge distribution there is a finite value of electric potential at the center of the sphere, at the surface of the sphere and also at a point outside 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 $\frac{q\rho}{3\varepsilon_0}$

Statement- $2:$ The electric field at a distance $r\,(r < R)$ from the center of the sphere is $\frac{\rho r}{3\varepsilon_0}$

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Answer: (c)

Statement 2: Gauss's law inside, $E \cdot 4\pi r^2 = \frac{\rho \frac{4}{3}\pi r^3}{\varepsilon_0} \Rightarrow E = \frac{\rho r}{3\varepsilon_0}$ . True.
Statement 1: $V_{centre} - V_{surface} = \int_0^R \frac{\rho r}{3\varepsilon_0}dr = \frac{\rho R^2}{6\varepsilon_0}$ .
So $\Delta U = \frac{q\rho R^2}{6\varepsilon_0}$ , not $\frac{q\rho}{3\varepsilon_0}$ (that is not even in energy units). False.
So the answer is (c) Statement 1 false, Statement 2 true.

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