Gauss’s Theorem
Gauss’s theorem, also known as Gauss’s law, states that the total electric flux passing through a closed surface is equal to \( \frac{1}{\varepsilon_0} \) times the total charge enclosed within that surface. Mathematically it is expressed as:
\[
\Phi = \oint \vec{E}\cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}
\]
where \( \vec{E} \) is the electric field, \( d\vec{A} \) is the small area element of the closed surface, \(Q_{enc}\) is the total charge enclosed, and \( \varepsilon_0 \) is the permittivity of free space.
Electric Field due to an Infinitely Long Charged Wire
Consider an infinitely long straight wire carrying a uniform linear charge density \( \lambda \). To apply Gauss’s theorem, imagine a cylindrical Gaussian surface of radius \( r \) and length \( l \) surrounding the wire. The electric field at every point on the curved surface of the cylinder is radial and has the same magnitude because of the symmetry of the charge distribution.
The total electric flux through the curved surface of the cylinder is given by:
\[
\Phi = E \times (2\pi r l)
\]
The charge enclosed inside the Gaussian surface is:
\[
Q_{enc} = \lambda l
\]
According to Gauss’s law:
\[
E(2\pi r l) = \frac{\lambda l}{\varepsilon_0}
\]
Solving for the electric field \(E\):
\[
E = \frac{\lambda}{2\pi \varepsilon_0 r}
\]
This shows that the electric field produced by an infinitely long charged wire decreases inversely with the distance from the wire.
Conclusion
Gauss’s theorem is a powerful tool for calculating electric fields in situations with high symmetry. Using Gauss’s law, the electric field at a distance \(r\) from an infinitely long straight charged wire is given by \(E = \frac{\lambda}{2\pi \varepsilon_0 r}\).