A very large and thick sheet of conducting material carries a non-uniform current density  $\overrightarrow{J}=J_0{\left(\frac{x}{d}\right)}^2\widehat{y}.$ The slab extends form x = –d to x = d, and is infinite in y and z.

Current Density in an Infinitely Large Thick Sheet

A very large and thick sheet of conducting material carries a non-uniform current density:

J = J₀ · (x/d)² in the 𝔎_y direction, where the slab extends from x = -d to x = +d and is infinite in y and z.

Total Current

The total current through the sheet is calculated by integrating the current density across the thickness:

I = ∫_-d^+d J₀ · (x/d)² dx

= J₀ ∫_-d^+d (x²/d²) dx

= J₀/d² × ∫_-d^+d x² dx

Evaluating the integral:

∫_-d^+d x² dx = [(x³/3)]_-d^+d = (d³/3) - (-d³/3) = 2d³/3

I = J₀ / d² × 2d³/3 = (2/3) J₀ d

Interpretation

• The current density is strongest at the sheet's edges (x = ±d) and zero at the center (x = 0).
• The total current per unit area is (2/3) J₀ d.