Ch 21: Electric Charge and Electric Field

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 21: Electric Charge and Electric Field

Problem 8b

Chapter 21, Problem 8b

Two small aluminum spheres, each having mass $0.0250$ kg, are separated by $80.0$ cm. How many electrons would have to be removed from one sphere and added to the other to cause an attractive force between the spheres of magnitude $1.00 \times 10^{4}$ N (roughly $1$ ton)? Assume that the spheres may be treated as point charges.

Verified step by step guidance

First, understand that the problem involves calculating the number of electrons needed to create a specific electrostatic force between two charged spheres. The force between two point charges is given by Coulomb's Law:

F = \frac{k q_{1} q_{2}}{r^{2}}

, where

k

is Coulomb's constant,

q

are the charges, and

r

is the separation distance.

Next, rearrange Coulomb's Law to solve for the product of the charges:

q_{1} q_{2} = \frac{F}{k} r^{2}

. Substitute the given values:

= 1.00 imes 10^4 \(\text{ N}\)

= 0.80 \(\text{ m}\)

, and

= 8.99 imes 10^9 \(\text{ N m}\)^2/\(\text{C}\)^2

Assume that the charge on one sphere is

q_{1} = q

and the charge on the other sphere is

q_{2} = - q

(since one sphere loses electrons and the other gains the same number). Thus,

q_{1} q_{2} = - q^{2}

Substitute

- q^{2}

into the equation from step 2 to find

q

q^{2} = \frac{F}{k} r^{2}

. Solve for

q

by taking the square root.

Finally, calculate the number of electrons,

n

, by dividing the charge

q

by the elementary charge

= 1.60 imes 10^{-19} \(\text{ C}\)

n = \frac{q}{e}

. This will give you the number of electrons that need to be transferred to achieve the desired force.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Coulomb's Law

Coulomb's Law describes the electrostatic interaction between charged particles. 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. The formula is F = k * |q1 * q2| / r^2, where k is Coulomb's constant, q1 and q2 are the charges, and r is the separation distance.

Charge Quantization

Charge quantization refers to the principle that charge comes in discrete units, specifically multiples of the elementary charge, e, which is the charge of a single electron (approximately 1.6 x 10^-19 coulombs). In this problem, the number of electrons transferred will determine the net charge on each sphere, affecting the electrostatic force between them.

Point Charge Approximation

The point charge approximation simplifies calculations by treating charged objects as if all their charge is concentrated at a single point. This is valid when the size of the charged objects is much smaller than the distance between them, allowing us to use Coulomb's Law directly without considering the distribution of charge across the object's surface.

Recommended video:

Guided course

08:35

Angular Momentum of a Point Mass

Related Practice

Textbook Question

If a proton and an electron are released when they are $2.0 \times 10^{- 10}$ m apart (a typical atomic distance), find the initial acceleration of each particle.

views

Textbook Question

Two small aluminum spheres, each having mass $0.0250$ kg, are separated by $80.0$ cm. How many electrons does each sphere contain? (The atomic mass of aluminum is $26.982$ g/mol, and its atomic number is $13$ .)

views

Textbook Question

Two small spheres spaced $20.0$ cm apart have equal charge. How many excess electrons must be present on each sphere if the magnitude of the force of repulsion between them is $3.33 \times 10^{- 21}$ N?

views

Textbook Question

Two small aluminum spheres, each having mass $0.0250$ kg, are separated by $80.0$ cm. What fraction of all the electrons in each sphere does this represent?

views

Textbook Question

Two small plastic spheres are given positive electric charges. When they are $15.0$ cm apart, the repulsive force between them has magnitude $0.220$ N. What is the charge on each sphere if the two charges are equal?

views

Textbook Question

views