Section 1.6: Line, Surface & Volume Charge Density
Charge density describes how charge is distributed in space. Depending on geometry, we define linear, surface, and volume charge densities.
Charge Densities:
- Line: \( \lambda = \frac{dq}{dl} \) [C/m]
- Surface: \( \sigma = \frac{dq}{dA} \) [C/m²]
- Volume: \( \rho = \frac{dq}{dV} \) [C/m³]
Infinitesimal charges \( dq = \lambda dl, \sigma dA, \rho dV \) are used in integration to compute forces or fields.
Example 1
A rod of length L has uniform linear charge density \( \lambda \). Find total charge and force on a point charge at distance d from the rod.
Total charge: \( Q = \lambda L \)
Force: \( \vec{F} = k_e q \int_0^L \frac{\lambda dx}{(x+d)^2} \hat{i} \)
Practice Problems
- Compute total charge on a circular loop with uniform linear density.
- Find charge on a square sheet with uniform surface density.
- Determine charge in a sphere with uniform volume density.
- Force on point charge near a uniformly charged rod.
- Integration of non-uniform density: \( \lambda(x) = \lambda_0 x \).