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Gauss Law

Gauss's law states that the net electric flux through a closed surface is directly proportional to the enclosed electric charge. It was formulated by Carl Friedrich Gauss in 1835 and relates the electric field at points on a closed surface to the net charge enclosed. The electric flux is defined as the electric field passing through an area multiplied by the area, perpendicular to the field. Gauss's law can also be expressed as the net flux through a surface, divided by the enclosed charge, being equal to a constant.

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96 views2 pages

Gauss Law

Gauss's law states that the net electric flux through a closed surface is directly proportional to the enclosed electric charge. It was formulated by Carl Friedrich Gauss in 1835 and relates the electric field at points on a closed surface to the net charge enclosed. The electric flux is defined as the electric field passing through an area multiplied by the area, perpendicular to the field. Gauss's law can also be expressed as the net flux through a surface, divided by the enclosed charge, being equal to a constant.

Uploaded by

Lorielle Oliva
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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Gauss’s law

Gauss

Gauss’s law states that the net flux of an electric field in a closed surface is directly proportional to the
enclosed electric charge. It is one of the four equations of Maxwell’s laws of electromagnetism. It was
initially formulated by Carl Friedrich Gauss in the year 1835 and relates the electric fields at the points
on a closed surface and the net charge enclosed by that surface.

The electric flux is defined as the electric field passing through a given area multiplied by the area of
the surface in a plane perpendicular to the field. Yet another statement of Gauss’s law states that the net
flux of a given electric field through a given surface, divided by the enclosed charge should be equal to
a constant.

Usually, a positive electric charge is supposed to generate a positive electric field. The law was
released in 1867 as part of a collection of work by the famous German mathematician, Carl Friedrich
Gauss.

Gauss Law Equation


Let us now study Gauss’s law through an integral equation. Gauss’s law in integral form is given
below:

∫E⋅dA=Q/ε0 ….. (1)

Where,

E is the electric field vector


Q is the enclosed electric charge
ε0 is the electric permittivity of free space
A is the outward pointing normal area vector
Flux is a measure of the strength of a field passing through a surface. Electric flux is defined as

Φ=∫E⋅dA …. (2)

We can understand the electric field as flux density. Gauss’s law implies that the net electric flux
through any given closed surface is zero unless the volume bounded by that surface contains a net
charge.

Gauss’s law for electric fields is most easily understood by neglecting electric displacement (d). In
matters, the dielectric permittivity may not be equal to the permittivity of free-space (i.e. ε≠ε0). In the
matter, the density of electric charges can be separated into a “free” charge density (ρf) and a
“bounded” charge density (ρb), such that:

Ρ = ρf + ρb

SAMPLE PROBLEMS

1. There are three charges q1, q2, and q3 having charge 6 C, 5 C and 3 C enclosed in a surface. Find
the total flux enclosed by the surface.

Answer: Total charge Q,

Q = q1 + q2 + q3
=6C+5C+3C
= 14 C

The total flux, ϕ = Q/ϵ0


ϕ = 14C / (8.854×10−12 F/m)
ϕ = 1.584 Nm2/C

Therefore, the total flux enclosed by the surface is 1.584 Nm2/C.


2. A uniform electric field of magnitude E = 100 N/C exists in the space in X-direction. Using the
Gauss theorem calculate the flux of this field through a plane square area of edge 10 cm placed in the
Y-Z plane. Take the normal along the positive X-axis to be positive.

Solution:

The flux Φ = ∫ E.cosθ ds.

As the normal to the area points along the electric field, θ = 0.

Also, E is uniform so, Φ = E.ΔS = (100 N/C) (0.10m)2 = 1 N-m2.

3. Determine the electric flux for a Gaussian surface that contains 100 million electrons.
Φ = q/εo
Φ = 100x10
6(1.6x10
-19)/8.85x10
-12
Φ = 1.8 Nm2/C

4. Find the flux through a spherical Gaussian surface of radius a = 1 m


surrounding a charge of 8.85 pC.
Solution. The flux thru the Gaussian surface is the charge located inside the
surface.

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