Question

A luminous object is separated from a screen by a distance $D$. What is the greatest focal length a lens could have to focus the object on the screen?

• A. $D$
• B. $4D$
• C. $D/4$
• D. $D/2$

Answer 1

The correct option is C $D/4$

Lens formula.

$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$

$\frac{1}{f}=\frac{1}{D-u}-\frac{1}{-u}$

$\frac{1}{f}=\frac{1}{0-u}+\frac{1}{u}$

$\begin{array}{cc}\frac{1}{f}& =\frac{u+0-u}{u(0-u)}\\ f& =\frac{u(0-u)}{D}-\left(1\right)\end{array}$

$u$ is varying

$f$ should be maximum

So $\frac{df}{du}=0$

$f=\frac{u(D-u)}{D}$

$\frac{df}{du}=\frac{1}{D}[\begin{array}{c}D-u+u(-1)]=0\\ D=2u\end{array}$

$\begin{array}{c}u=D{/}_2\end{array}$

Substitute in eq (1)

$f=\frac{\frac{D}{2}(D-\frac{D}{2})}{D}$

$f_{max}=\frac{D}{4}$

So option $C$ is correct.