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Walkway illuminance from 4750 cd lamps

Short answer

Illuminance E = I h / d³. (a)(i) Directly under lamp: E = 4750/6² = 131.94 lux. (a)(ii) 16 m away: E ≈ 5.71 lux. (b) Midway (8 m from each of two lamps): each contributes 28.5 lux, total 57.0 lux.

A walkway is illuminated by 250W lamps each having a luminous intensity of 4750 candela in all directions below the horizontal. Each lamp is installed at a height of 6m and the distance between them is 16 meters. Calculate the illuminance contributed by each lamp: a) (i) directly underneath, (ii) 16 meters from the base, b) The total illuminance at Midway between the base of each lamp post.

Each lamp has I = 4750 cd in all directions below the horizontal, mounting height h = 6 m, spacing 16 m.

For a point at horizontal distance x from the base:

d = √(h² + x²)
cos θ = h / d
E = I cos θ / d² = I h / d³

(a)(i) Directly underneath (x = 0)

E = I / h² = 4750 / 6² = 4750 / 36
  = 131.94 lux

(a)(ii) 16 m from the base (x = 16 m)

d² = 6² + 16² = 36 + 256 = 292
d  = 17.09 m
cos θ = 6 / 17.09 = 0.351
E = 4750 × 6 / 292^1.5
  = 28 500 / 4989.8
  = 5.71 lux

(b) Total illuminance midway between two lamp posts (x = 8 m)

Each of the two adjacent lamps is 8 m away horizontally:

d² = 6² + 8² = 100,  d = 10 m
cos θ = 6 / 10 = 0.6
E (one lamp) = 4750 × 0.6 / 100 = 28.5 lux
E (total)    = 2 × 28.5         = 57.0 lux

(The next lamps on either side, 24 m away, add only about 1.88 lux each, so they are normally neglected.)

   L1                    L2
   ●──────── 16 m ───────●
   │6 m                  │
 ──┴──────────┬──────────┴──
   P1       Pmid(8 m)    P2

Answer: (a)(i) 131.94 lux, (a)(ii) 5.71 lux per lamp; (b) total at midway = 57.0 lux (28.5 lux from each lamp).

✓ Answer checked by Claude Opus 5.5 (high effort)

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