Ch 21: Electric Charge and Electric Field

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 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.

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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.

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