25. Electric Potential

Draw the electric field that corresponds to the equipotential surfaces shown in the following figure. Note that the potential is decreasing in the upward direction.

A very large plastic sheet carries a uniform charge density of $-6.00$ nC/m² on one face. As you move away from the sheet along a line perpendicular to it, does the potential increase or decrease? How do you know, without doing any calculations? Does your answer depend on where you choose the reference point for potential?

Suppose the charge on the outer sphere is not $-q$ but a negative charge of different magnitude, say $-Q$. Show that the answers for parts (b) and (c) are the same as before but the answer for part (d) is different. Note: Part (a) asked to calculate the potential $V(r)$ for (i) $r < r_a$; (ii) $r_a < r < r_b$; (iii) $r > r_b$. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take $V$ to be zero when $r$ is infinite. Part (b) asked to show that the potential of the inner sphere with respect to the outer is $V_ab=q/(4πϵ_0 ) (1/r_a -1/r_b)$. Part (c) asked to use $E_r=-∂V/∂r=(-∂/∂r) (1/(4πϵ_0 ) q/r)=[1/(4πϵ_0 )](q/r^2)$ and the result from part (a) to show that the electric field at any point between the spheres has magnitude $E(r)=[V_ab/(1/r_a -1/r_b )](1/r^2)$. Part (d) asked to use $E_r = [1/(4πϵ_0 )](q/r^2)$ and the result from part (a) to find the electric field at a point outside the larger sphere at a distance $r$ from the center, where $r > r_b$.

A metal sphere with radius $r_a$ is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius $r_b$. There is charge $+q$ on the inner sphere and charge $-q$ on the outer spherical shell. Use $E_r=\left[1/\right(4\pi {ϵ}_0\left)\right]\left(q/r^2\right)$ and the result from part (a) to find the electric field at a point outside the larger sphere at a distance $r$ from the center, where $r>r_b$. Note: Part (a) asked to calculate the potential $V(r)$ for (i) $r < r_a$; (ii) $r_a < r < r_b$; (iii) $r>r_b$. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take $V$ to be zero when $r$ is infinite..

A metal sphere with radius $r_a$ is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius $r_b$. There is charge $+q$ on the inner sphere and charge $-q$ on the outer spherical shell. Use $E_r=-\partial V/\partial r=(-\partial /\partial r)\left(1/\right(4\pi {ϵ}_0\left)q/r\right)=\left[1/\right(4\pi {ϵ}_0\left)\right]\left(q/r^2\right)$ and the result from part (a) to show that the electric field at any point between the spheres has magnitude $E\left(r\right)=\left[V_{a}b/\right(1/r_a-1/r_b\left)\right]\left(1/r^2\right)$. Note: Part (a) asked to calculate the potential $V(r)$ for (i) $r < r_a$; (ii) $r_a < r < r_b$; (iii) $r > r_b$. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take $V$ to be zero when $r$ is infinite..