Ch 26: Potential and Field

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 26: Potential and Field

Problem 64

Chapter 26, Problem 64

An isolated 5.0 μF parallel-plate capacitor has 4.0 mC of charge. An external force changes the distance between the electrodes until the capacitance is 2.0 μF. How much work is done by the external force?

Verified step by step guidance

Start by recalling the formula for the energy stored in a capacitor: \( U = \frac{1}{2} \frac{Q^2}{C} \), where \( U \) is the energy, \( Q \) is the charge, and \( C \) is the capacitance. Since the capacitor is isolated, the charge \( Q \) remains constant.

Calculate the initial energy stored in the capacitor using the initial capacitance \( C_1 = 5.0 \ \mu\text{F} \) and the charge \( Q = 4.0 \ \text{mC} \). Substitute these values into the energy formula: \( U_1 = \frac{1}{2} \frac{Q^2}{C_1} \).

Next, calculate the final energy stored in the capacitor after the capacitance is reduced to \( C_2 = 2.0 \ \mu\text{F} \). Use the same formula: \( U_2 = \frac{1}{2} \frac{Q^2}{C_2} \).

Determine the work done by the external force by finding the change in energy of the capacitor. Since the external force increases the distance between the plates, the capacitance decreases, and the energy increases. The work done is given by \( W = U_2 - U_1 \).

Substitute the expressions for \( U_1 \) and \( U_2 \) into the work formula to find \( W \). Simplify the expression to calculate the work done by the external force.

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Key Concepts

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

Capacitance

Capacitance is the ability of a capacitor to store charge per unit voltage, measured in farads (F). For a parallel-plate capacitor, capacitance (C) is determined by the formula C = ε₀(A/d), where ε₀ is the permittivity of free space, A is the area of the plates, and d is the distance between them. Changes in the distance between the plates affect capacitance, which is crucial for understanding how the capacitor behaves when external forces are applied.

Work Done by External Force

The work done by an external force on a capacitor can be calculated using the formula W = ΔU, where ΔU is the change in electrical potential energy. The potential energy (U) stored in a capacitor is given by U = 1/2 C V², where C is capacitance and V is voltage. As the capacitance changes due to the external force, the work done can be determined by finding the difference in potential energy before and after the change.

Charge Conservation

In an isolated capacitor, the total charge remains constant even if the capacitance changes. This principle of charge conservation means that when the capacitance decreases, the voltage across the capacitor must increase to maintain the same charge (Q = C × V). Understanding this relationship is essential for calculating the new voltage and subsequently the work done by the external force when the distance between the capacitor plates is altered.

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05:43

Conservation of Charge

Related Practice

Textbook Question

Initially, the switch in FIGURE P26.61 is in position A and capacitors C₂ and C₃ are uncharged. Then the switch is flipped to position B. Afterward, the voltage across C₁ is 4.0 V. What is the emf of the battery?

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Textbook Question

High-frequency signals are often transmitted along a coaxial cable, such as the one shown in FIGURE P26.68. For example, the cable TV hookup coming into your home is a coaxial cable. The signal is carried on a wire of radius R₁ while the outer conductor of radius R₂ is grounded (i.e., at V=0 V). An insulating material fills the space between them, and an insulating plastic coating goes around the outside. Evaluate the capacitance per meter of a cable having R₁=0.50 mm and R₂=3.0 mm.

Textbook Question

Six identical capacitors with capacitance C are connected as shown in FIGURE P26.59. What is the potential difference between points a and b?

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Textbook Question

Capacitors C₁ = 10 μF and C₂ = 20 μF are each charged to 10 V, then disconnected from the battery without changing the charge on the capacitor plates. The two capacitors are then connected in parallel, with the positive plate of C₁ connected to the negative plate of C₂ and vice versa. Afterward, what are the charge on and the potential difference across each capacitor?

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Textbook Question

You've built a device that uses the energy from a rapidly discharged capacitor to launch the capacitor straight up. One capacitor, with a mass of 3.5 g, is launched to a height of 1.6 m after having been charged to 100 V. What is its capacitance in μF?

Textbook Question

The label rubbed off one of the capacitors you are using to build a circuit. To find out its capacitance, you place it in series with a 10 μF capacitor and connect them to a 9.0 V battery. Using your voltmeter, you measure 6.0 V across the unknown capacitor. What is the unknown capacitor's capacitance?

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