Question

Derive the formula for the electric field due to a uniformly charged straight wire of infinite length by using Gauss' law.

Hint:

The electric field around an infinite charged wire decreases inversely with the distance from the wire. Gauss' law is a powerful tool to calculate electric fields with symmetry.

Answer 1

Step 1: Gauss' Law.
Gauss' law states that the electric flux \( \Phi_E \) through a closed surface is proportional to the total charge enclosed within the surface: \[ \Phi_E = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \] where \( \mathbf{E} \) is the electric field, \( d\mathbf{A} \) is the differential area vector, \( Q_{\text{enc}} \) is the total charge enclosed, and \( \epsilon_0 \) is the permittivity of free space.
Step 2: Symmetry of the Problem.
Consider a uniformly charged straight wire of infinite length. The electric field produced by this wire is radially symmetric and points directly outward (or inward if the charge is negative). The field depends only on the distance \( r \) from the wire. We will use a cylindrical Gaussian surface with radius \( r \) and length \( L \) centered on the wire. The total charge enclosed by the Gaussian surface is \( \lambda L \), where \( \lambda \) is the linear charge density (charge per unit length) of the wire.
Step 3: Electric Flux Calculation.
Since the electric field is radial and uniform over the surface of the cylinder, the flux through the curved surface of the cylinder is: \[ \Phi_E = E \times (2 \pi r L) \] where \( E \) is the magnitude of the electric field and \( 2 \pi r L \) is the surface area of the curved side of the cylindrical Gaussian surface.
Step 4: Apply Gauss’ Law.
Using Gauss' law, the total electric flux is also equal to the charge enclosed divided by \( \epsilon_0 \): \[ \frac{Q_{\text{enc}}}{\epsilon_0} = \frac{\lambda L}{\epsilon_0} \] Equating the two expressions for electric flux, we get: \[ E \times (2 \pi r L) = \frac{\lambda L}{\epsilon_0} \]
Step 5: Solve for the Electric Field.
Canceling \( L \) from both sides, we get the electric field at distance \( r \) from the wire: \[ E = \frac{\lambda}{2 \pi r \epsilon_0} \]
Step 6: Conclusion.
Thus, the electric field due to a uniformly charged straight wire of infinite length is: \[ E = \frac{\lambda}{2 \pi r \epsilon_0} \]