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 47

Chapter 22, Problem 47

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

Verified step by step guidance

Step 1: Identify the charges and their positions in FIGURE P22.47. Note the magnitudes of the charges, their coordinates, and the distances between them. The charge at the bottom is 1.0 nC, and we need to calculate the net force acting on it due to other charges in the system.

Step 2: Use Coulomb's law to calculate the force between two point charges. Coulomb's law is given by:

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

, where

k

is the Coulomb constant (

8.99 \times 10 9^{}

N·m²/C²),

q

and

q'

are the charges, and

r

is the distance between them.

Step 3: Break the forces into components. For each pair of charges, calculate the x and y components of the force using trigonometry. The x-component is given by

F \times \cos (θ)

, and the y-component is given by

F \times \sin (θ)

, where

θ

is the angle between the force vector and the horizontal axis.

Step 4: Sum up the components of the forces from all charges acting on the 1.0 nC charge. Add the x-components together to find the net force in the x-direction, and add the y-components together to find the net force in the y-direction.

Step 5: Express the net force in component form as

(F

_x,F_y), where

F

_x and

F

_y are the net forces in the x and y directions, respectively.

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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 (F) is directly proportional to the product of the magnitudes of the charges (q1 and q2) and inversely proportional to the square of the distance (r) between them. The formula is F = k * (|q1 * q2|) / r², where k is Coulomb's constant. This law is fundamental for calculating the forces acting on charged particles.

Vector Components

In physics, forces are often represented as vectors, which have both magnitude and direction. To analyze forces in a two-dimensional space, it is useful to break them down into their components along the x and y axes. This allows for easier calculations and understanding of the net force acting on an object. The components can be found using trigonometric functions based on the angle of the force vector.

Superposition Principle

The Superposition Principle states that when multiple forces act on a charge, the total force is the vector sum of all individual forces. This principle is crucial in electrostatics, as it allows us to calculate the net force on a charge by considering the contributions from all other charges in the system. By applying this principle, one can determine the resultant force acting on the 1.0 nC charge by adding the forces due to other nearby charges.

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Related Practice

Textbook Question

In a simple model of the hydrogen atom, the electron moves in a circular orbit of radius 0.053 nm around a stationary proton. How many revolutions per second does the electron make?

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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

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.

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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?

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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.

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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?