Electric Field of Line Charge

An infinite line of charge produces an electric field that decreases as $1/r$ rather than $1/r^2$. This is a classic application of Gauss's Law.

Setup

  • Infinite line with uniform linear charge density $\lambda$ (C/m)
  • Want to find $E$ at distance $r$ from the line

Derivation Using Gauss's Law

Gaussian Surface

Cylindrical surface of radius $r$ and length $L$, coaxial with the line charge.

Analysis

Curved surface:

  • $\vec{E}$ is parallel to $\vec{A}$ (radially outward)
  • $\vec{E}$ has constant magnitude at distance $r$
  • Flux: $\Phi_{curved} = E(2\pi r L)$

End caps:

  • $\vec{E}$ is perpendicular to $\vec{A}$
  • Flux: $\Phi_{ends} = 0$

Total flux: $$\Phi_E = E(2\pi r L)$$

Enclosed Charge

$$Q_{enclosed} = \lambda L$$

Apply Gauss's Law

$$\Phi_E = \frac{Q_{enclosed}}{\varepsilon_0}$$

$$E(2\pi r L) = \frac{\lambda L}{\varepsilon_0}$$

$$E = \frac{\lambda}{2\pi r \varepsilon_0}$$

Final Result

$$E = \frac{\lambda}{2\pi \varepsilon_0 r}$$

Or using $k = \frac{1}{4\pi\varepsilon_0}$:

$$E = \frac{2k\lambda}{r}$$

Key Properties

Property Value
Direction Radially outward (perpendicular to line)
Magnitude $E \propto \frac{1}{r}$
Dependence on $\lambda$ Linear: $E \propto \lambda$

Comparison with Point Charge

Source Field Dependence
Point charge $E \propto \frac{1}{r^2}$
Line charge $E \propto \frac{1}{r}$
Plane sheet $E = $ constant

The field falls off more slowly because the source extends infinitely.

Example Calculation

Problem: A long straight wire has charge per unit length $\lambda = 3.00 \times 10^{-12}$ C/m. At what distance is $E = 0.600$ N/C?

Solution: $$r = \frac{\lambda}{2\pi \varepsilon_0 E} = \frac{3.00 \times 10^{-12}}{2\pi (8.85 \times 10^{-12})(0.600)}$$

$$r = 0.090 \text{ m} = 9.0 \text{ cm}$$

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