A very long, straight wire has charge per unit length 3.20×10−103.20\$$times10^{-10} C/m. At $$what distance from the wire is the electric field magnitude equal to 2.502.50 N/C?

Ch 21: Electric Charge and Electric Field

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 21: Electric Charge and Electric Field

Problem 50

Chapter 21, Problem 50

A very long, straight wire has charge per unit length $3.20 \times 10^{- 10}$ C/m. At what distance from the wire is the electric field magnitude equal to $2.50$ N/C?

Verified step by step guidance

First, recognize that the problem involves a long, straight wire with a uniform linear charge density, which suggests using Gauss's Law to find the electric field around the wire.

Gauss's Law states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space:

Φ = \frac{Q}{ε 0}

For a long, straight wire, consider a cylindrical Gaussian surface coaxial with the wire. The electric field is radial and constant over the curved surface of the cylinder.

The electric field magnitude

E

at a distance

r

from the wire is given by:

E = \frac{λ}{2 π r ε 0}

, where

λ

is the linear charge density.

Set the given electric field magnitude

2.50

N/C equal to the expression for

E

and solve for

r

r = \frac{λ}{2 π ε 0 E}

. Substitute the values for

λ

ε 0

, and

E

to find

r

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Key Concepts

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

Linear Charge Density

Linear charge density is the amount of electric charge per unit length along a wire or line. It is denoted by λ and is measured in coulombs per meter (C/m). In this problem, the wire has a linear charge density of 3.20 * 10^-10 C/m, which is crucial for calculating the electric field around the wire.

Electric Field due to a Line of Charge

The electric field generated by a long, straight wire with uniform linear charge density can be calculated using Gauss's Law. The electric field (E) at a distance (r) from the wire is given by E = (λ / (2πε₀r)), where ε₀ is the permittivity of free space. This formula helps determine the distance at which the electric field magnitude is specified.

Permittivity of Free Space

Permittivity of free space, denoted as ε₀, is a fundamental physical constant that describes how electric fields interact with the vacuum. Its value is approximately 8.85 * 10^-12 C²/(N·m²). This constant is essential in calculating the electric field around charged objects, such as the wire in this problem, using Gauss's Law.

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A point charge $q_{1} = - 4.00$ nC is at the point $x equals 0.600$ m, $y equals 0.800$ m, and a second point charge $q sub 2 equals positive 6.00$ nC is at the point $x equals 0.600$ m, $y equals 0$ . Calculate the magnitude and direction of the net electric field at the origin due to these two point charges.

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A point charge is placed at each corner of a square with side length $a$ . All charges have magnitude $q$ . Two of the charges are positive and two are negative (Fig. E $21.42$ ). What is the direction of the net electric field at the center of the square due to the four charges, and what is its magnitude in terms of $q$ and $a$ ?

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Point charges $q_{1} = - 4.5$ nC and $q sub 2 equals positive 4.5$ nC are separated by $3.1$ mm, forming an electric dipole. The charges are in a uniform electric field whose direction makes an angle of $36.9$ ° with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude $7.2 \times 10^{- 9}$ Nm?

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Textbook Question

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$ . Find the orientations of the dipole for which the torque on the dipole is zero.

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Textbook Question

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$ . Which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small rotation away from the equilibrium position and see what happens.) Note: Part (a) asked to find the orientations of the dipole for which the torque on the dipole is zero.

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Textbook Question

A charge of $negative 6.50$ nC is spread uniformly over the surface of one face of a nonconducting disk of radius $1.25$ cm. Why is the field in part (a) stronger than the field in part (b)? Why is the field in part (c) the strongest of the three fields? Note: Part (a) asked to find the magnitude and direction of the electric field this disk produces at a point $P$ on the axis of the disk a distance of $2.00$ cm from its center. Part (b) asked to find the magnitude and direction of the electric field at point $P$ , supposing that the charge were all pushed away from the center and distributed uniformly on the outer rim of the disk. Part (c) asked to find the magnitude and direction of the electric field at point $P$ if the charge is all brought to the center of the disk.

A very long, straight wire has charge per unit length 3.20×10−103.20\(\times\)10^{-10} C/m. At what distance from the wire is the electric field magnitude equal to 2.502.50 N/C?

Verified video answer for a similar problem:

Key Concepts

Linear Charge Density

Electric Field due to a Line of Charge

Permittivity of Free Space

A very long, straight wire has charge per unit length $3.20 \times 10^{- 10}$ C/m. At what distance from the wire is the electric field magnitude equal to $2.50$ N/C?