Question

The electric field in a region is given by `vec"E"` = 5 `hatk`N/C. Calculate the electric flux Through a square of side 10.0 cm in the following cases

1. The square is along the XY plane
2. The square is along XZ plane
3. The normal to the square makes an angle of 45° with the Z axis.

Answer 1

Given: `vec"E"` = 5 `hatk` N/C, |E| = 5 N/C, l = 10 cm = 10 × 10⁻² m = 10⁻¹ m, A = l² = 10⁻²m²

To find: Electric flux in three cases (Φ₁), (Φ₂), (Φ₃)

(i) Given the plane of Area is along XY plane. hence the direction of its normal `(vec"dS")` vector will be along Z axis.

The Angle between `vec"E"` and `"d"vec"S"` is zero i.e. θ = 0°.

∴ Φ = EA cos θ

∴ Φ₁ = 5 × 10⁻² cos 0
∴ Φ₁ = 5 × 10⁻² V m

(ii) Plane of Area is along XZ plane. Hence the direction of its normal `(vec"dS")` vector will be along Y axis.

∴ θ = 90°

∴ Φ = EA cos θ

∴ Φ₂ = 5 × 10⁻² cos 90°

∴ Φ₂ = 0 V m

(iii) When normal to the square makes an angle of 45° with the Z axis.

∴ θ = 45°

∴ Φ = EA cos θ

∴ Φ₃ = 5 × 10⁻² cos 45°

∴ Φ₃ = 3.5 × 10⁻² V m

Electric flux in the above-mentioned cases are 5 × 10⁻² V m, 0 V m, and 3.5 × 10⁻² V m.