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 33

Chapter 26, Problem 33

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?

Verified step by step guidance

Step 1: Recall the formula for the energy density of an electric field, which is given by:

u = (1/2)ε₀E²

, where

u

is the energy density,

ε₀

is the permittivity of free space (

8.85 × 10⁻¹² \, \(\text{F/m}\)

), and

E

is the electric field strength.

Step 2: Calculate the volume of the region where the energy is stored. The region is a cube with side length

2.0 \, \(\text{cm}\)

, so the volume is:

V = (2.0 \, \(\text{cm}\))³ = (0.02 \, \(\text{m}\))³

Step 3: Use the relationship between the total energy stored in the electric field and the energy density:

U = uV

, where

U

is the total energy stored. Rearrange to find the energy density:

u = U/V

. Substitute

U = 50 \, \(\text{pJ}\) = 50 × 10⁻¹² \, \(\text{J}\)

and the volume calculated in Step 2.

Step 4: Substitute the expression for energy density

u

into the formula

u = (1/2)ε₀E²

. Rearrange to solve for the electric field strength

E

E = \(\sqrt{(2u/ε₀)}\)

Step 5: Substitute the value of

u

from Step 3 and

ε₀ = 8.85 × 10⁻¹² \, \(\text{F/m}\)

into the equation from Step 4 to calculate the electric field strength

E

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Key Concepts

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

Electric Field Energy

The energy stored in an electric field can be expressed using the formula U = 1/2 * ε * E^2 * V, where U is the energy, ε is the permittivity of the medium, E is the electric field strength, and V is the volume of the region. In this case, the energy stored is given as 50 pJ, and the volume can be calculated from the dimensions provided.

Volume of a Cubic Region

The volume of a cube is calculated using the formula V = side^3. For a cube with each side measuring 2.0 cm, the volume can be converted to cubic meters for consistency in SI units. This volume is essential for determining the electric field strength when combined with the energy stored in the field.

Electric Field Strength

Electric field strength (E) is defined as the force (F) per unit charge (q) experienced by a positive test charge placed in the field, expressed as E = F/q. It can also be derived from the energy stored in the electric field and the volume, allowing us to solve for E when the energy and volume are known.

Recommended video:

Guided course

03:16

Intro to Electric Fields

Related Practice

Textbook Question

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?

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Textbook Question

The electric field in a region of space is E_x = −1000x² V/m, where x is in meters. What is the potential difference between x_i = −20 cm and x_f = 30 cm?

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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