Question

A student is studying a book placed near the edge of a circular table of radius R. A point source of light is suspended directly above the centre of the table. What should be the height of the source above the table so as to produce maximum illuminance at the position of the book?

Answer 1

Let the height of the source be h and the luminous intensity in the normal direction be I₀.

So, illuminance on the book (E) is given by,

\[E = \frac{I_0 \cos\theta}{r^2}\]

From the figure we get

Cosθ = \[\frac{h}{r}\]

On substituting the value of Cosθ, we get

\[E = \frac{I_0 h}{r^3}\]

But r = \[\sqrt{R^2 + h^2}\]

\[\therefore E=\frac{I_0 h}{\left( r^2 + h^2 \right)^\frac{3}{2}}\]

For maximum illuminance,

\[\frac{dE}{dH} = 0\]

\[\frac{dE}{dH} = \frac{I_0 \left[ \left( R^2 + h^2 \right)^\frac{3}{2} - \frac{3}{2}h \times \left( R^2 + h^2 \right)^\frac{1}{2} \times 2h \right]}{\left( R^2 + h^2 \right)^3} = 0\]

\[\Rightarrow  \left( R^2 + h^2 \right)^\frac{1}{2} \left[ R^2 + h^2 - 3 h^2 \right] = 0\]

\[ \Rightarrow  R^2  - 2 h^2  = 0\]

\[ \Rightarrow h = \frac{R}{\sqrt{2}}\]