Ch 22: Electric Charges and Forces

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 22: Electric Charges and Forces

Problem 22a

Chapter 22, Problem 22a

A 2.0 g plastic bead charged to −4.0 nC and a 4.0 g glass bead charged to +8.0 nC are 2.0 cm apart and free to move. What are the accelerations of the plastic bead?

Verified step by step guidance

Identify the forces acting on the beads. The force between the two charged beads is given by Coulomb's law:

F = k_e \(\frac{{|q_1 q_2|}\)}{{r^2}}

, where

k_e

is the Coulomb constant (

8.99 \(\times\) 10^9 \, \(\text{N·m}\)^2/\(\text{C}\)^2

q_1

and

q_2

are the charges of the beads, and

r

is the distance between them.

Substitute the given values into Coulomb's law. Here,

q_1 = -4.0 \, \(\text{nC}\) = -4.0 \(\times\) 10^{-9} \, \(\text{C}\)

q_2 = +8.0 \, \(\text{nC}\) = 8.0 \(\times\) 10^{-9} \, \(\text{C}\)

, and

r = 2.0 \, \(\text{cm}\) = 0.020 \, \(\text{m}\)

. Calculate the magnitude of the force

F

Determine the acceleration of the plastic bead using Newton's second law:

a = \(\frac{F}{m}\)

. The mass of the plastic bead is

m = 2.0 \, \(\text{g}\) = 0.002 \, \(\text{kg}\)

. Use the force calculated in the previous step to find the acceleration of the plastic bead.

Determine the acceleration of the glass bead using the same formula,

a = \(\frac{F}{m}\)

. The mass of the glass bead is

m = 4.0 \, \(\text{g}\) = 0.004 \, \(\text{kg}\)

. Use the same force (since the force is equal and opposite on both beads) to calculate the acceleration of the glass bead.

Summarize the results. The plastic bead and the glass bead experience equal and opposite forces due to Coulomb's law, but their accelerations differ because of their different masses. Ensure the directions of the accelerations are consistent with the attractive nature of the force (the plastic bead accelerates toward the glass bead, and vice versa).

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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 force between two charged objects. It states that the force 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 is fundamental in calculating the forces acting on charged particles, which is essential for determining their accelerations.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed by the formula F = ma, where F is the net force, m is the mass, and a is the acceleration. Understanding this law is crucial for calculating how the forces from the charged beads affect their motion.

Mass and Charge

Mass and charge are fundamental properties of matter that influence how objects interact with forces. In this scenario, the mass of each bead affects how much they accelerate in response to the electrostatic forces calculated using Coulomb's Law. The charge determines the strength and direction of the forces acting on each bead, which is vital for solving the problem of their accelerations.

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