Question

State Gauss’s Theorem. Use it to derive the electric field intensity due to an infinitely long straight charged wire.

Hint:

Use Gauss’s law when symmetry exists: Spherical $\rightarrow$ sphere, Cylindrical $\rightarrow$ cylinder, Planar $\rightarrow$ Gaussian pillbox.

Answer 1

Gauss’s Theorem:
Gauss’s Theorem states that the total electric flux through a closed surface is equal to 1/ε₀ times the total charge enclosed within that surface.

Mathematically,
∮ E · dA = Q_enclosed / ε₀

Where E is the electric field, dA is the area element of the closed surface, Q_enclosed is the total charge inside the surface, and ε₀ is the permittivity of free space.

Derivation of Electric Field due to an Infinitely Long Straight Charged Wire:

Consider an infinitely long straight wire having a uniform linear charge density λ (charge per unit length). To find the electric field at a distance r from the wire, we use Gauss’s law.

Step 1: Choose a Suitable Gaussian Surface.
Due to cylindrical symmetry, we choose a cylindrical Gaussian surface of radius r and length L, coaxial with the charged wire.

Step 2: Determine the Direction of Electric Field.
Because of symmetry, the electric field at every point on the curved surface of the cylinder is directed radially outward and has the same magnitude E. The field is perpendicular to the curved surface and parallel to the end caps.

Step 3: Calculate the Electric Flux.
Electric flux through the curved surface:
Flux = E × (Curved surface area)
Curved surface area of cylinder = 2πrL

So,
Total flux = E × 2πrL

There is no flux through the flat ends of the cylinder because the electric field is parallel to those surfaces.

Step 4: Apply Gauss’s Law.
According to Gauss’s law:
E × 2πrL = Q_enclosed / ε₀

Charge enclosed within length L of the wire:
Q_enclosed = λL

So,
E × 2πrL = λL / ε₀

Cancel L from both sides:
E × 2πr = λ / ε₀

Step 5: Final Expression for Electric Field.
E = λ / (2π ε₀ r)

Conclusion:
The electric field intensity at a distance r from an infinitely long straight charged wire is
E = λ / (2π ε₀ r)

The field decreases inversely with distance (1/r) and is directed radially outward if the charge is positive, and inward if the charge is negative.