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Line charge and sheet charge, find D

Short answer

D from the line charge is 4a_x C/m². From the sheet, D = −10a_x C/m². The total flux density at (0.5,0,0) is the sum: D = −6a_x C/m².

Along the z-axis there is a uniform line of charge with ρ_L = 4π C/m and in the x = 1 plane there is a surface charge with ρ_s = 20 C/m². Find the electric flux density at (0.5, 0, 0).

Superpose the flux densities of the line charge (z-axis) and the sheet (plane x = 1) at the point (0.5, 0, 0).

Line charge on the z-axis, ρ_L = 4π C/m

D = ρ_L/(2πρ) a_ρ. The point is at ρ = 0.5 m on the +x side, so a_ρ = a_x.

D_L = 4π/(2π × 0.5) a_x = 4 a_x  C/m²

Sheet at x = 1, ρ_s = 20 C/m²

D = (ρ_s/2) a_n, with a_n pointing from the sheet to the point. The point (x = 0.5) is on the side x < 1, so a_n = −a_x.

D_S = (20/2)(−a_x) = −10 a_x  C/m²

Total

D = D_L + D_S = 4 a_x − 10 a_x = −6 a_x  C/m²

Answer: D = −6 a_x C/m² at (0.5, 0, 0).

✓ Answer checked by Claude Opus 5.5 (high effort)

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