- 2068 Baisakh · 5 marks
Express W field in cylindrical, spherical
Short answer
W = (x-y) a_y. In cylindrical: W = ρ(cosφ - sinφ)(sinφ a_ρ + cosφ a_φ). In spherical: W = r sinθ (cosφ - sinφ)(sinθ sinφ a_r + cosθ sinφ a_θ + cosφ a_φ).
Express the vector field W = (x − y) a_y in cylindrical and spherical co-ordinates.
Given: W = (x − y) a_y.
Only W_y = x − y is non-zero. Use x = ρ cosφ, y = ρ sinφ (cylindrical) and x = r sinθ cosφ, y = r sinθ sinφ (spherical). The dot products of a_y with the unit vectors are: a_y · a_ρ = sinφ, a_y · a_φ = cosφ, a_y · a_z = 0; and a_y · a_r = sinθ sinφ, a_y · a_θ = cosθ sinφ.
Cylindrical
x − y = ρ (cosφ − sinφ)
W_ρ = (x − y) sinφ = ρ (cosφ − sinφ) sinφ
W_φ = (x − y) cosφ = ρ (cosφ − sinφ) cosφ
W_z = 0
W = ρ(cosφ − sinφ) [ sinφ a_ρ + cosφ a_φ ]
Spherical
x − y = r sinθ (cosφ − sinφ)
W_r = (x − y) sinθ sinφ = r sin²θ (cosφ − sinφ) sinφ
W_θ = (x − y) cosθ sinφ = r sinθ cosθ (cosφ − sinφ) sinφ
W_φ = (x − y) cosφ = r sinθ (cosφ − sinφ) cosφ
W = r sinθ (cosφ − sinφ) [ sinθ sinφ a_r + cosθ sinφ a_θ + cosφ a_φ ]
Check (cylindrical): W_ρ² + W_φ² = ρ²(cosφ − sinφ)² = (x − y)² = |W|² ✓
✓ Answer checked by Claude Opus 5.5 (high effort)
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