A 1.0-mm-diameter ball bearing has 2.0×109 excess electrons. What is the ball bearing's potential?

Ch 25: The Electric Potential

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 25: The Electric Potential

Problem 28

Chapter 25, Problem 28

A 1.0-mm-diameter ball bearing has 2.0×10⁹ excess electrons. What is the ball bearing's potential?

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Determine the charge on the ball bearing by multiplying the number of excess electrons by the elementary charge. Use the formula:

q = n \cdot e

, where

n

is the number of excess electrons and

e

is the elementary charge (approximately

1.6 \times 10^{-19}

C).

Calculate the radius of the ball bearing from its diameter. The radius is half the diameter:

r = \frac{d}{2}

, where

d

is the diameter (1.0 mm or

1.0 \times 10^{-3}

m).

Use the formula for the electric potential on the surface of a sphere:

V = \frac{k \cdot q}{r}

, where

k

is Coulomb's constant (

8.99 \times 10^{9}

N·m²/C²),

q

is the charge, and

r

is the radius of the ball bearing.

Substitute the values for

q

r

, and

k

into the formula for

V

. Ensure all units are consistent (e.g., meters for radius and Coulombs for charge).

Simplify the expression to find the electric potential

V

on the surface of the ball bearing. This will give the final result in volts (V).

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Key Concepts

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

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. Charges can be positive or negative, with electrons carrying a negative charge. The total charge of an object is the sum of its positive and negative charges, and in this case, the ball bearing has an excess of electrons, indicating a net negative charge.

Electric Potential

Electric potential, often referred to as voltage, is the amount of electric potential energy per unit charge at a point in an electric field. It is measured in volts (V) and represents the work done to move a charge from a reference point to a specific point in the field. The potential of the ball bearing can be calculated using the formula V = k * Q / r, where k is Coulomb's constant, Q is the total charge, and r is the distance from the charge.

Coulomb's Law

Coulomb's Law describes the force between two charged objects and is fundamental in electrostatics. It states that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law helps in understanding how the excess electrons on the ball bearing influence its electric potential and the forces it may exert on other charges.

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