Electric Field Due To An Infinitely Long Charged Wire

In this article, we will learn to calculate electric field due to infinite line charge or electric field due to an infinitely long straight, uniformly charged wire.

Table of Contents:

What is Electric Field?

An electric field is defined as the electric force per unit charge. It is given as:

\(\begin{array}{l}\vec{E} = \vec{F}/Q\end{array} \)

Where,

  • E is the electric field intensity
  • F is the force on the charge “Q.”
  • Q is the charge

Variations in the magnetic field or the electric charges cause electric fields. Volt per metre (V/m) is the SI unit of the electric field.

What is Gaussian Surface?

A closed surface in a three-dimensional space whose flux of a vector field is calculated, which can either be the magnetic field, the electric field, or the gravitational field, is known as the Gaussian Surface. Let us learn how to calculate the electric field due to infinite line charges.

Gaussian surface formula

Electric Field Due To An Infinitely Long Straight Uniformly Charged Wire

Let us learn how to calculate the electric field due to infinite line charges. Consider an infinitely long straight, uniformly charged wire. Let the linear charge density of this wire be λ. P is the point that is located at a perpendicular distance from the wire. The distance between point P and the wire is r.

Electric field due to an infinitely long charged wire

The wire is considered to be a cylindrical Gaussian surface. This is because to determine the electric field E at point P, Gauss law is used.

The surface area of the curved part is given as:

S = 2πrl

The total charge enclosed by the Gaussian surface is given as:

q = λl

The electric flux through the end surfaces of the cylindrical Gaussian surface is given as:

Φ1 = 0

The electric flux through the curved surface of the cylindrical Gaussian surface is given as:

Φ2 = E cosθ.s

Φ2 = E x 1 x 2πrl

The total electric flux is given as:

Φ = Φ1 + Φ2

Φ = 0 + E cosθ.s

Φ2 = 2πrlE (eq. 1)

From Gauss law, we know that

\(\begin{array}{l}\varphi =\frac{q}{\epsilon _{o}}=\frac{\lambda l}{\epsilon_{o}} (eq.2)\end{array} \)

From eq 1. And eq 2

\(\begin{array}{l}2πrlE = \frac{\lambda l}{\epsilon_{o}}\end{array} \)
\(\begin{array}{l}E=\frac{1}{2\pi \epsilon _{o}}\frac{\lambda }{r}\end{array} \)

Therefore, the above equation is the electric field due to an infinitely long straight uniformly charged wire.

Learn about the characteristics of electrical force with the help of the video below.

Hope you have learned about an electric field due to an infinite line charge or an electric field due to an infinitely long straight, uniformly charged wire. Stay tuned with BYJU’S to learn more about other concepts.

Frequently Asked Questions – FAQs

Q1

What is the Gaussian surface of a sphere?

The following equation is the Gaussian surface of a sphere:

\(\begin{array}{l}E=\frac{Q_{A}}{4\pi \epsilon _{o}r^{2}}\end{array} \)

Q2

What is the Gaussian surface of a cylinder?

The following equation is the Gaussian surface of a cylinder:

\(\begin{array}{l}E=\frac{\lambda }{2\pi \epsilon _{o}r}\end{array} \)

Q3

Can an electric field be zero?

Yes, an electric field can be zero. For, unlike charges, the electric field is zero outside when the magnitude of the charge is small. The electric field is zero along the line joining the two charges for like charges.

Q4

At which point is the electric field the weakest?

We know that the electric field is proportional to the density of the field lines. Therefore, the electric field is the weakest when the field lines are far.

Q5

What direction do electric fields travel?

The direction of the electric fields is the same as the direction of the electrical force acting on the positive charge.

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