Ch 23: Electric Potential

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 23: Electric Potential

Problem 40

Chapter 23, Problem 40

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?

Verified step by step guidance

First, understand that the potential at the center of a charged sphere is the same as the potential at its surface due to the symmetry of the sphere. The formula for the electric potential V at the surface of a sphere is given by:

V = \frac{k}{Q} r

, where k is Coulomb's constant, Q is the charge, and r is the radius of the sphere.

Convert the diameter of the sphere to radius. Since the diameter is 25.0 cm, the radius r is half of that, which is 12.5 cm or 0.125 m.

Rearrange the formula to solve for the charge Q:

Q = \frac{V}{r} k

. Substitute the given potential V = 3.75 kV (or 3750 V), the radius r = 0.125 m, and Coulomb's constant k = 8.99 x 10^9 N m²/C² into the equation.

Calculate the charge Q using the rearranged formula. This will give you the excess charge required to achieve the given potential at the center of the sphere.

For part (b), note that the potential at the surface of the sphere relative to infinity is the same as the potential at the center, due to the properties of conductors and spherical symmetry. Therefore, the potential at the surface is also 3.75 kV.

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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 is the amount of electric potential energy per unit charge at a point in a field. It is measured in volts (V) and represents the work done to move a charge from a reference point, often infinity, to a specific point in the field. For a charged sphere, the potential at any point inside is the same as on its surface.

Capacitance of a Sphere

The capacitance of a sphere is a measure of its ability to store charge per unit potential difference. For an isolated sphere, the capacitance C is given by C = 4πε₀R, where R is the radius of the sphere and ε₀ is the permittivity of free space. This concept helps relate the charge on the sphere to its potential.

Relation between Charge and Potential

The potential V of a charged sphere is related to its charge Q and radius R by the formula V = Q / (4πε₀R). This relationship allows us to calculate the excess charge needed to achieve a specific potential. It also implies that the potential is uniform across the sphere's surface and inside it.

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Related Practice

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. Show that the potential of the inner sphere with respect to the outer is $V_{a} b = q / (4 π ϵ_{0}) (1 / r_{a} - 1 / r_{b})$ .

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

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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Textbook Question

Two large, parallel conducting plates carrying opposite charges of equal magnitude are separated by $2.20$ cm. What is the potential difference between the two plates?

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Textbook Question

Two large, parallel conducting plates carrying opposite charges of equal magnitude are separated by $2.20$ cm. The surface charge density for each plate has magnitude $47.0$ nC/m^2. If the separation between the plates is doubled while the surface charge density is kept constant at the given value, what happens to the magnitude of the electric field and to the potential difference?

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How much excess charge must be placed on a copper sphere 25.025.0 cm in diameter so that the potential of its center, rela­tive to infinity, is 3.753.75 kV? What is the potential of the sphere's surface relative to infinity?

Verified video answer for a similar problem:

Key Concepts

Electric Potential

Capacitance of a Sphere

Relation between Charge and Potential

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?