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 44

Chapter 22, Problem 44

What is the force F on the 8.0 nC charge in FIGURE P22.44? Give your answer as a magnitude and an angle measured cw or ccw (specify which) from the +x-axis.

Verified step by step guidance

Identify the charges and their positions in FIGURE P22.44. Note the 8.0 nC charge and the other charges that exert forces on it. Assign coordinates to each charge for clarity.

Use Coulomb's Law to calculate the force between the 8.0 nC charge and each of the other charges. Coulomb's Law is given by:

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

, where

k

is the Coulomb constant,

q_{1}

and

q_{2}

are the charges, and

r

is the distance between them.

Break each force into its x and y components. Use trigonometry to find the components:

F_{x} = F \cos θ

and

F_{y} = F \sin θ

, where

F

is the magnitude of the force and

θ

is the angle relative to the x-axis.

Sum up all the x-components and y-components of the forces to find the net force components:

F_{x} = \sum F_{x}

and

F_{y} = \sum F_{y}

Calculate the magnitude of the net force using the Pythagorean theorem:

F = \sqrt{{F_{x}}^{2} + {F_{y}}^{2}}

. Determine the angle of the net force relative to the +x-axis using:

θ = \tan -^{1} \frac{F_{y}}{F_{x}}

. Specify whether the angle is measured clockwise (cw) or counterclockwise (ccw) from the +x-axis.

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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 for calculating the force acting on a charge due to other charges in its vicinity.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In the context of forces, each force can be represented as a vector with both magnitude and direction. The overall force on a charge is found by adding the individual force vectors, taking into account their directions, which can be expressed in terms of angles.

Coordinate System

A coordinate system provides a framework for specifying the position and direction of forces in a two-dimensional plane. In this problem, the +x-axis is typically used as a reference for measuring angles. Understanding how to convert forces into components along the x and y axes is crucial for accurately determining the magnitude and direction of the resultant force on the charge.

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Related Practice

Textbook Question

FIGURE P22.52 shows three charges and the net force on charge −q. Charge Q is some multiple α of q. What is α?

Textbook Question

What is the force F on the 1.0 nC charge at the bottom in FIGURE P22.47? Give your answer in component form.

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Textbook Question

Two 1.0 g spheres are charged equally and placed 2.0 cm apart. When released, they begin to accelerate at 150 m/s². What is the magnitude of the charge on each sphere?

Textbook Question

A +2.0 nC charge is at the origin and a −4.0 nC charge is at x = 1.0 cm. Would the net force be zero for an electron placed at the same position? Explain.

Textbook Question

A 3.00-cm-long spring has a small plastic bead glued to each end. Charging each bead to −25 nC expands the spring by 0.50 cm. What is the value of the spring constant?

Textbook Question

Objects A and B are both positively charged. Both have a mass of 100 g, but A has twice the charge of B. When A and B are placed 10 cm apart, B experiences an electric force of 0.45 N. What is the charge on A?