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 ?

Explanation

To solve the problem of finding the height \( h \) of the point source of light above the table that produces maximum illuminance at the position of the book, we can follow these steps:

### Step 1: Understand the Geometry
We have a circular table with radius \( R \) and a point source of light suspended directly above the center of the table at height \( h \). The book is placed at the edge of the table.

### Step 2: Define the Illuminance Formula
The illuminance \( E \) at a point on the table is given by the formula:
\[ E = \frac{I_0 \cos \theta}{d^2} \]
where \( I_0 \) is the intensity of the light source, \( \theta \) is the angle between the light ray and the normal to the surface, and \( d \) is the distance from the light source to the point on the table.

### Step 3: Relate \( \cos \theta \) to Height and Radius
From the geometry of the situation, we can relate \( \cos \theta \) to the height \( h \) and the radius \( R \) using:
\[ \cos \theta = \frac{h}{\sqrt{R^2 + h^2}} \]
This represents the vertical component of the light intensity.

### Step 4: Calculate the Distance \( d \)
The distance \( d \) from the light source to the edge of the table (where the book is) can be calculated using the Pythagorean theorem:
\[ d = \sqrt{R^2 + h^2} \]

### Step 5: Substitute into the Illuminance Formula
Substituting \( \cos \theta \) and \( d \) into the illuminance formula gives:
\[ E = \frac{I_0 \cdot \frac{h}{\sqrt{R^2 + h^2}}}{R^2 + h^2} \]
This simplifies to:
\[ E = \frac{I_0 h}{(R^2 + h^2)^{3/2}} \]

### Step 6: Maximize the Illuminance
To find the height \( h \) that maximizes \( E \), we need to take the derivative of \( E \) with respect to \( h \) and set it to zero:
\[ \frac{dE}{dh} = 0 \]
Using the quotient rule for differentiation, we differentiate \( E \).

### Step 7: Solve the Derivative Equation
After differentiating and simplifying, we arrive at the equation:
\[ R^2 - 2h^2 = 0 \]
This implies:
\[ R^2 = 2h^2 \]
Solving for \( h \) gives:
\[ h = \frac{R}{\sqrt{2}} \]

### Final Answer
Thus, the height \( h \) of the source above the table for maximum illuminance at the position of the book is:
\[ h = \frac{R}{\sqrt{2}} \]

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