A metal sphere with radius ra r_ara is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius rbr_brb. There is charge +q+q+q on the inner sphere and charge −q-q−q on the outer spherical shell. Show that the potential of the inner sphere with respect to the outer is Vab=q/(4πϵ0)(1/ra−1/rb)V_ab=q/(4πϵ_0 ) (1/r_a -1/r_b).

Ch 23: Electric Potential

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 23: Electric Potential

Problem 41b

Chapter 23, Problem 41b

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 $V_{a} b = q / (4 π ϵ_{0}) (1 / r_{a} - 1 / r_{b})$ .

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Step 1: To solve part (a), we need to calculate the potential V(r) in three regions: (i) inside the inner sphere (r < r_a), (ii) between the inner sphere and the outer shell (r_a < r < r_b), and (iii) outside the outer shell (r > r_b). Use the formula for the potential due to a point charge, V(r) = q/(4πϵ_0 r), and consider the contributions from both the inner sphere and the outer shell.

Step 2: For part (b), calculate the potential difference between the inner sphere and the outer shell, V_ab. Use the formula V_ab = q/(4πϵ_0) (1/r_a - 1/r_b), which represents the potential difference due to the charge distribution on the spheres.

Step 3: In part (c), use the expression for the electric field E_r = -∂V/∂r = [1/(4πϵ_0)](q/r^2) and the result from part (a) to show that the electric field between the spheres has magnitude E(r) = [V_ab/(1/r_a - 1/r_b)](1/r^2). This involves differentiating the potential function with respect to r.

Step 4: For part (d), find the electric field at a point outside the larger sphere (r > r_b) using E_r = [1/(4πϵ_0)](q/r^2). This requires applying the formula for the electric field due to a point charge and considering the charge distribution on the spheres.

Step 5: In part (e), consider the case where the charge on the outer sphere is -Q instead of -q. Show that the potential difference V_ab and the electric field between the spheres remain unchanged, but the electric field outside the larger sphere is affected by the change in charge magnitude. This involves re-evaluating the potential and electric field expressions with the new charge configuration.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Potential

Electric potential, V, is the work done per unit charge in bringing a positive test charge from infinity to a point in space. It is a scalar quantity and is measured in volts. In this problem, the potential at a point is the sum of potentials due to the inner and outer spheres, and it is zero at infinity.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It is essential for calculating the electric field in symmetric charge distributions, such as spherical shells. Here, it helps determine the electric field between and outside the spheres based on the charge distribution.

Electric Field

The electric field, E, is a vector field representing the force per unit charge exerted on a test charge at any point in space. It is derived from the gradient of the electric potential, E = -∂V/∂r. In this problem, the field is calculated between and outside the spheres using the potential difference and charge distribution.

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Textbook Question

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?

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Textbook Question

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 $V (r)$ 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.

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Textbook Question

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 $E_{r} = [1 / (4 π ϵ_{0})] (q / r^{2})$ 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 $r > r_{b}$ . 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..

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Textbook Question

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 $E_{r} = - \partial V / \partial r = (- \partial / \partial r) (1 / (4 π ϵ_{0}) q / r) = [1 / (4 π ϵ_{0})] (q / r^{2})$ and the result from part (a) to show that the electric field at any point between the spheres has magnitude $E (r) = [V_{a} b / (1 / r_{a} - 1 / r_{b})] (1 / r^{2})$ . 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..

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Textbook Question

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 $1.0$ $μ$ 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?

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