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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
All textbooksKnight Calc 5th EditionCh 22: Electric Charges and ForcesProblem 68
Chapter 22, Problem 68

An electric field E→=200,000i^\(\overrightarrow{E}\)=200,000\(\hat{i}\) N/C causes the point charge in FIGURE P22.68 to hang at an angle. What is θ?
Diagram showing a point charge of 25 nC and 2.0 g hanging at angle θ in an electric field of 200,000 N/C.

Verified step by step guidance
1
Identify the forces acting on the point charge: The forces include the gravitational force \( F_g = mg \), the electric force \( F_e = qE \), and the tension \( T \) in the string. The tension has components in both the horizontal and vertical directions.
Write the force balance equations: In the vertical direction, the tension's vertical component balances the gravitational force, \( T \sin(\theta) = mg \). In the horizontal direction, the tension's horizontal component balances the electric force, \( T \cos(\theta) = qE \).
Divide the horizontal force equation by the vertical force equation to eliminate \( T \): \( \frac{T \cos(\theta)}{T \sin(\theta)} = \frac{qE}{mg} \). Simplify this to \( \tan(\theta) = \frac{qE}{mg} \).
Solve for \( \theta \): Use the inverse tangent function, \( \theta = \arctan\left(\frac{qE}{mg}\right) \).
Substitute the known values into the equation: Use the given electric field \( E = 200,000 \ \text{N/C} \), the charge \( q \), the mass \( m \), and the acceleration due to gravity \( g = 9.8 \ \text{m/s}^2 \) to calculate \( \theta \).

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Key Concepts

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

Electric Field

An electric field is a region around a charged particle where other charged particles experience a force. It is represented by the symbol E and is measured in newtons per coulomb (N/C). The strength and direction of the electric field depend on the charge creating it and the distance from that charge.
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Point Charge

A point charge is an idealized model of a charged object that has negligible size compared to the distances involved in the problem. It is characterized by its charge (positive or negative) and creates an electric field around it. The behavior of point charges in an electric field can be analyzed using Coulomb's law and the principles of electrostatics.
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Equilibrium of Forces

In the context of the problem, equilibrium of forces refers to the condition where the net force acting on the point charge is zero. This occurs when the gravitational force acting downward is balanced by the electric force acting upward, allowing the charge to hang at an angle θ. Analyzing these forces helps determine the angle at which the charge remains in equilibrium.
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Related Practice
Textbook Question

Space explorers discover an 8.7×1017 kg asteroid that happens to have a positive charge of 4400 C. They would like to place their 3.3×105 kg spaceship in orbit around the asteroid. Interestingly, the solar wind has given their spaceship a charge of −1.2C. What speed must their spaceship have to achieve a 7500-km-diameter circular orbit?

Textbook Question

Three 1.0 nC charges are placed as shown in FIGURE P22.66. Each of these charges creates an electric field E at a point 3.0 cm in front of the middle charge. What are the three fields E₁, E₂, and E₃ created by the three charges? Write your answer for each as a vector in component form.

Textbook Question

The identical small spheres shown in FIGURE P22.64 are charged to +100 nC and −100 nC. They hang as shown in a 100,000 N/C electric field. What is the mass of each sphere?

Textbook Question

A 5.0 g ball charged to 1.5 μC is tied to a 25-cm-long string. It swings at 250 rpm in a horizontal circle around a stationary ball charged to −2.5 μC. What is the tension in the string?

Textbook Question

A 10.0 nC charge is located at position (x, y)=(1.0 cm, 2.0 cm). At what (x, y) position(s) is the electric field (21,600i^−28,800j^)(21,600\(\hat{i}\)-28,800\(\hat{j}\)) N/C?

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Textbook Question

Starting from rest, how long does it take an electron to move 1.0 cm in a steady electric field of magnitude 100 N/C?

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