Ch 26: Potential and Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 26: Potential and Field

Problema 33

Capitolo 26, Problema 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?

Guida verificata passo dopo passo

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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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.

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Percorso guidato

03:16

Intro to Electric Fields

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.

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Percorso guidato

05:21

Volume Thermal Expansion

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.

Video consigliato:

Percorso guidato

03:16

Intro to Electric Fields

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