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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 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 $$\frac{qP}{3\varepsilon_0}$$.
Statement 2: The electric field at a distance $$r\ (r < R)$$ from the centre of the sphere is $$\frac{\rho r}{3\varepsilon_0}$$
Evaluating Statement 2 using Gauss's law for $$r < R$$:
$$E \cdot 4\pi r^2 = \frac{\rho \cdot \frac{4}{3}\pi r^3}{\varepsilon_0} \implies E = \frac{\rho r}{3\varepsilon_0}$$
Evaluating Statement 1 using the relationship between potential difference and electric field:
$$\Delta V = V_{\text{surface}} - V_{\text{centre}} = -\int_0^R E \, dr = -\int_0^R \frac{\rho r}{3\varepsilon_0} \, dr = -\frac{\rho R^2}{6\varepsilon_0}$$
$$\Delta U = q \Delta V = -\frac{q\rho R^2}{6\varepsilon_0}$$
Comparing calculated change in potential energy to the statement's expression $$\frac{q\rho}{3\varepsilon_0}$$:
$$\Delta U \neq \frac{q\rho}{3\varepsilon_0} \implies \text{Statement 1 is false}$$
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