11. A 20 cm x 30 cm sheet of paper is illuminated by a candle 10 metres away. The
total light power falling on it is X. The paper is now brought to within 10 cm of the
candle. The total light power falling on it is now Y. What is the ratio Y:X?
a. Below 10000:1
b. Exactly 10000:1
c. Above 10000:1
d. Cannot be determined from the given information
Candle Paper Illumination Ratio: Inverse Square Law Explained
The ratio Y:X for light power on a 20 cm × 30 cm paper sheet moving from 10 meters to 10 cm from a candle exceeds 10,000:1 when using the standard inverse square law approximation, but exact solid angle calculations show it’s actually below this value due to the close-distance limitation.
Core Physics Principle
Light from a point source like a candle follows the inverse square law, where illuminance (power per unit area) decreases with the square of distance: \(E \propto \frac{1}{d^2}\).- For total power on a surface, multiply by area, but when the surface subtends a small solid angle \(\Omega \approx \frac{A}{d^2}\), total power \(P = I \cdot \Omega\) (I is source intensity). At 10 m, the paper’s small angular size (∼1.7°) validates this, yielding \(\Omega_1 \approx 0.0006\) steradians.
Detailed Calculation
At 10 m (\(d_1 = 10\) m), approximate \(\Omega_1 = \frac{0.2 \times 0.3}{10^2} = 6 \times 10^{-4}\) sr, so \(X \propto 6 \times 10^{-4}\).At 10 cm (\(d_2 = 0.1\) m), simple \(\Omega_2 = \frac{0.06}{0.01} = 6\) sr gives ratio \(\frac{Y}{X} = \left(\frac{10}{0.1}\right)^2 = 10,000\). Exact solid angle \(\Omega = 4 \arctan\left(\frac{(a/2)(b/2)}{d \sqrt{d^2 + (a/2)^2 + (b/2)^2}}\right)\) yields \(\Omega_2 \approx 2.52\) sr (not 6 sr, as it nears hemisphere limit of \(2\pi \approx 6.28\) sr), so \(\frac{Y}{X} \approx 4194 < 10,000\).
Option Analysis
- a. Below 10000:1 – Correct for precise physics; close proximity (angular size ∼171°) violates point-source/small-angle assumptions, capping power below naive prediction.
- b. Exactly 10000:1 – Matches simple \(\left(\frac{d_1}{d_2}\right)^2\), valid only if paper stays small relative to distance (true at 10 m, false at 10 cm).
- c. Above 10000:1 – Impossible; maximum power is half total candle output (\(2\pi\) sr hemisphere), far below 10,000X.
- d. Cannot be determined – Incorrect; candle approximates point source at 10 m, and solid angle formula determines ratio definitively.
Exam Context
This JGEEBILS/CSIR NET-style question tests inverse square law limits for finite surfaces.[web:6] Simple approximation suggests b, but advanced analysis (solid angle) favors a, emphasizing when approximations fail in optics/illumination engineering.
