Ch 25: The Electric Potential

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 25: The Electric Potential

Problem 64a

Chapter 25, Problem 64a

The potential 1.0 cm from the surface of a metal sphere is 8000 V. The potential 3.0 cm from the surface is 4000 V. What is the radius of the sphere?

Verified step by step guidance

Recognize that the potential outside a charged sphere behaves like the potential of a point charge, given by the formula:

V = \frac{k Q}{r}

, where

V

is the potential,

k

is Coulomb's constant,

Q

is the charge, and

r

is the distance from the center of the sphere.

Set up two equations for the given potentials at distances 1.0 cm and 3.0 cm from the surface of the sphere. Let the radius of the sphere be

R

. The distances from the center of the sphere are

R + 1.0

cm and

R + 3.0

cm, respectively.

Write the equations for the potentials:

8000 = \frac{k Q}{R + 1.0}

and

4000 = \frac{k Q}{R + 3.0}

Divide the first equation by the second to eliminate

k Q

. This gives:

\frac{8000}{4000} = \frac{R + 3.0}{R + 1.0}

. Simplify this to:

2 = \frac{R + 3.0}{R + 1.0}

Solve the simplified equation for

R

. Cross-multiply to get:

2 (R + 1.0) = R + 3.0

. Expand and simplify to isolate

R

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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, measured in volts (V), is the amount of electric potential energy per unit charge at a point in an electric field. It indicates how much work would be done to move a charge from a reference point to a specific point in the field. In this problem, the electric potential values at different distances from the sphere's surface are crucial for determining the sphere's radius.

Coulomb's Law

Coulomb's Law describes the force between two charged objects and is fundamental in electrostatics. It states that the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This law helps in understanding how the electric potential varies with distance from a charged object, such as the metal sphere in this question.

Spherical Symmetry in Electric Fields

Spherical symmetry in electric fields implies that the electric field and potential around a uniformly charged sphere behave uniformly in all directions. For a charged sphere, the electric potential outside the sphere can be treated as if all the charge were concentrated at the center. This concept is essential for calculating the radius of the sphere based on the given potential values at specified distances from its surface.

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