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 64

Chapter 22, Problem 64

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?

Verified step by step guidance

Step 1: Analyze the forces acting on the spheres. Each sphere experiences three forces: (1) the gravitational force $ F_g = m g $, where $ m $ is the mass of the sphere and $ g $ is the acceleration due to gravity, (2) the electric force $ F_e = q E $, where $ q $ is the charge on the sphere and $ E $ is the electric field strength, and (3) the tension force $ T $ in the string, which has both vertical and horizontal components.

Step 2: Break the tension force $ T $ into components. The vertical component $ T_y $ balances the gravitational force $ F_g $, and the horizontal component $ T_x $ balances the electric force $ F_e $. Use the relationships $ T_y = T \cos\theta $ and $ T_x = T \sin\theta $, where $ \theta $ is the angle the string makes with the vertical.

Step 3: Write the equilibrium conditions. For vertical equilibrium: $ T_y = F_g $, which gives $ T \cos\theta = m g $. For horizontal equilibrium: $ T_x = F_e $, which gives $ T \sin\theta = q E $.

Step 4: Divide the horizontal equilibrium equation by the vertical equilibrium equation to eliminate $ T $: $ \frac{T \sin\theta}{T \cos\theta} = \frac{q E}{m g} $. Simplify using $ \tan\theta $: $ \tan\theta = \frac{q E}{m g} $. Rearrange to solve for $ m $: $ m = \frac{q E}{g \tan\theta} $.

Step 5: Substitute the known values into the equation. Use $ q = 100 \, \text{nC} = 100 \times 10^{-9} \text{C} $, $ E = 100,000 \, \text{N/C} $, $ g = 9.8 \, \text{m/s}^2 $, and the given angle $ \theta $ (from the diagram in FIGURE P22.64). Calculate $ \tan\theta $ using the geometry of the setup, and then substitute all values to find $ m $.

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Key Concepts

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

Electric Force

Electric force is the interaction between charged objects, described by Coulomb's law. In this scenario, the charged spheres experience forces due to their respective charges in the presence of an external electric field. The force acting on a charged object in an electric field can be calculated using the formula F = qE, where F is the force, q is the charge, and E is the electric field strength.

Weight and Mass

Weight is the force exerted by gravity on an object, calculated as W = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.81 m/s² on Earth). To find the mass of the spheres, we need to relate the electric force acting on them to their weight, as they are in equilibrium under the influence of both forces.

Equilibrium of Forces

In this context, equilibrium refers to the state where the net force acting on the spheres is zero. This means that the upward electric force must balance the downward gravitational force (weight) acting on each sphere. By setting the electric force equal to the weight, we can solve for the mass of the spheres, allowing us to understand how the charges and the electric field influence their behavior.

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Related Practice

Textbook Question

Two equal point charges 2.5 cm apart, both initially neutral, are being charged at the rate of 5.0 nC/s. At what rate (N/s) is the force between them increasing 1.0 s after charging begins?

Textbook Question

An electric field $E right arrow equals 200 comma 000 i hat$ N/C causes the point charge in FIGURE P22.68 to hang at an angle. What is θ?

Textbook Question

An electric dipole consists of two opposite charges $\pm q$ separated by a small distance $s$ . The product $p = q s$ is called the dipole moment. Figure P $22.61$ shows an electric dipole perpendicular to an electric field $E$ . Find an expression in terms of $p$ and $E$ for the magnitude of the torque that the electric field exerts on the dipole.

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Textbook Question

You have a lightweight spring whose unstretched length is 4.0 cm. First, you attach one end of the spring to the ceiling and hang a 1.0 g mass from it. This stretches the spring to a length of 5.0 cm. You then attach two small plastic beads to the opposite ends of the spring, lay the spring on a frictionless table, and give each plastic bead the same charge. This stretches the spring to a length of 4.5 cm. What is the magnitude of the charge (in nC) on each bead?

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

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 $open paren 21 comma 600 i hat minus 28 comma 800 j hat close paren$ N/C?

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