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. Show that the potential of the inner sphere with respect to the outer is $V_{a}b=q/\left(4\pi {ϵ}_0\right)(1/r_a-1/r_b)$.

How much excess charge must be placed on a copper sphere $25.0$ cm in diameter so that the potential of its center, rela­tive to infinity, is $3.75$ kV? What is the potential of the sphere's surface relative to infinity?

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. Calculate the potential $V\left(r\right)$ for (i) ; (ii) ; (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 $E_r=\left[1/\right(4\pi {ϵ}_0\left)\right]\left(q/r^2\right)$ 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..

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 $r > r_b$.

Certain sharks can detect an electric field as weak as $1.0$ $\mu$V/m. To grasp how weak this field is, if you wanted to produce it between two parallel metal plates by connecting an ordinary $1.5$­V AA battery across these plates, how far apart would the plates have to be?