Question

The diagram shows a material formed into a sphere of radius R. The diagram shows the magnitude of the electric field E as a function of distance r from the center of the sphere. Which of the following can be concluded from this graph?
The diagram shows a material formed into a sphere of radius R. The diagram shows the magnitude of the electric field E as a function of distance r from the center of the sphere. Which of the following can be concluded from this graph?

• A. The sphere is made of metal
• B. The sphere is an insulator with uniform volume charge density
• C. The sphere is an insulator with volume charge density that increases as $r$ increases
• D. The sphere is an insulator with volume charge density that decreases as $r$ increases
• E. None of these conclusions is certain based on the graph

Answer 1

Answer: (B)

The sphere is an insulator with uniform volume charge density

For an insulated sphere charged uniformly, the charge inside sphere or radius $r<R$ is given by $\rho \left(\frac{4}{3}\pi r^3\right)$.

Hence electric field at a distance $r$ can be found by applying gauss law,

$E\left(4\pi r^2\right)=\frac{\rho \left(\frac{4}{3}\pi r^3\right)}{{ϵ}_0}$

$⟹E\propto r$

Hence curve upto $R$ is linear.

The electric field out of the sphere can be found by applying gauss law again.

$E\left(4\pi r^2\right)=\frac{\rho \left(\frac{4}{3}\pi R^3\right)}{{ϵ}_0}$

$⟹E\propto \frac{1}{r^2}$

Hence correct answer is option B.