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Lamp suspension height and illumination at offset

Short answer

Lamp height = 1.58 m for 100 lux directly below (h = √(I/E)). Illuminance at 1.2 m from vertical axis = 50.54 lux using cosine law (E = I·h/d³).

The illumination at a point on a working plane directly below the lamp is to be 100 lumen/m². The lamp gives 250 c.p. uniformly below the lamp horizontal plane. Determine the height at which the lamp is suspended. Also find illumination at a point on the working table 1.2m away from the vertical axis of the lamp.

Given: luminous intensity I = 250 cd (c.p.) in all directions below the horizontal; illuminance directly below the lamp E₀ = 100 lux (lm/m²).

Height of the lamp

For a point directly below the lamp, θ = 0 and distance = h:

E₀ = I / h²
h  = √(I / E₀) = √(250 / 100) = √2.5 = 1.581 m

Illuminance at 1.2 m from the vertical axis

By the cosine law, E = (I / d²) × cos θ, where d is the distance from lamp to point and cos θ = h/d.

d     = √(h² + x²) = √(2.5 + 1.2²) = √3.94 = 1.985 m
cos θ = h/d = 1.581/1.985 = 0.7966   (θ = 37.2°)
E     = I × h / d³
      = 250 × 1.581 / 1.985³
      = 395.3 / 7.821
      = 50.54 lux

Answer: lamp height = 1.58 m above the working table; illuminance at 1.2 m from the axis = 50.54 lux (lm/m²).

✓ Answer checked by Claude Opus 5.5 (high effort)

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