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 32

Chapter 26, Problem 32

A 2.0-cm-diameter parallel-plate capacitor with a spacing of 0.50 mm is charged to 200 V. What are (a) the total energy stored in the electric field and (b) the energy density?

Verified step by step guidance

Step 1: Calculate the area of one plate of the capacitor. The diameter is given as 2.0 cm, so the radius is 1.0 cm (0.01 m). Use the formula for the area of a circle: A = πr².

Step 2: Determine the capacitance of the parallel-plate capacitor using the formula C = (ε₀A) / d, where ε₀ is the permittivity of free space (8.85 × 10⁻¹² F/m), A is the area of the plates, and d is the separation between the plates (0.50 mm = 0.0005 m).

Step 3: Calculate the total energy stored in the capacitor using the formula U = (1/2)CV², where C is the capacitance and V is the voltage (200 V).

Step 4: Compute the energy density (u), which is the energy per unit volume, using the formula u = U / Volume. The volume of the capacitor is given by the product of the area of the plates and the separation distance: Volume = A × d.

Step 5: Combine the results from the previous steps to express the total energy stored (U) and the energy density (u) in terms of the given values and constants.

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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. It is defined as C = Q/V, where C is capacitance, Q is the charge stored, and V is the voltage across the plates. For parallel-plate capacitors, capacitance can also be calculated using 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 separation between them.

Energy Stored in a Capacitor

The energy (U) stored in a capacitor is given by the formula U = 1/2 CV², where C is the capacitance and V is the voltage. This equation shows that the energy stored is proportional to the square of the voltage and the capacitance. Understanding this relationship is crucial for calculating the total energy stored in the electric field of the capacitor.

Energy Density

Energy density (u) in an electric field is defined as the energy stored per unit volume. For a capacitor, it can be calculated using the formula u = 1/2 εE², where ε is the permittivity of the material between the plates and E is the electric field strength. This concept helps in understanding how energy is distributed within the electric field of the capacitor.

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Intro to Density

Related Practice

Textbook Question

50 pJ of energy is stored in a 2.0 cm×2.0 cm×2.0 cm region of uniform electric field. What is the electric field strength?

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Textbook Question

A 6 μF capacitor, a 10 μF capacitor, and a 16 μF capacitor are connected in series. What is their equivalent capacitance?

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Textbook Question

The electric field in a region of space is E_x=5000x V/m , where x is in meters. Find an expression for the potential V at position x. As a reference, let V=0 V at the origin.

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Textbook Question

What is the equivalent capacitance of the three capacitors in FIGURE EX26.27?

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Textbook Question

The electric field in a region of space is Ex = −1000x^2 V/m, where x is in meters. Graph Ex versus x over the region −1 m ≤ x ≤1 m.

Textbook Question

You need a capacitance of 50 μF, but you don't happen to have a 50 μF capacitor. You do have a 75 μF capacitor. What additional capacitor do you need to produce a total capacitance of 50 μF? Should you join the two capacitors in parallel or in series?

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