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NVAEasyJEE 2026Optical Instruments (Microscope, Telescope)

JEE Physics 2026 Question with Solution

A collimated beam of light of diameter 2 mm2 \, \text{mm} is propagating along the xx-axis. The beam is required to be expanded into a collimated beam of diameter 14 mm14 \, \text{mm} using a system of two convex lenses. If the first lens has focal length 40 mm40 \, \text{mm}, then the focal length of the second lens is _____ mm\text{mm}.

Answer

Correct answer:280

Step-by-step solution

Standard Method

Given: Input beam diameter is Din=2 mmD_{in} = 2 \, \text{mm}, output beam diameter is Dout=14 mmD_{out} = 14 \, \text{mm}, and focal length of the first convex lens is f1=40 mmf_1 = 40 \, \text{mm}.

Find: The focal length f2f_2 of the second convex lens.

A beam expander made of two convex lenses works like a Keplerian telescope. For such a system, the diameter magnification is equal to the ratio of focal lengths.

M=DoutDin=f2f1M = \frac{D_{out}}{D_{in}} = \frac{f_2}{f_1}

Substitute the given values:

142=f240\frac{14}{2} = \frac{f_2}{40}7=f2407 = \frac{f_2}{40}f2=7×40=280 mmf_2 = 7 \times 40 = 280 \, \text{mm}

Therefore, the focal length of the second lens is 280 mm280 \, \text{mm}.

Ratio Trick

Given: The beam diameter changes from 2 mm2 \, \text{mm} to 14 mm14 \, \text{mm}, and f1=40 mmf_1 = 40 \, \text{mm}.

Find: The value of f2f_2.

For a beam expander, the expansion factor is the same as the focal length ratio. Since the beam diameter increases by

142=7\frac{14}{2} = 7

the second focal length must also be 77 times the first focal length.

f2=7×40=280 mmf_2 = 7 \times 40 = 280 \, \text{mm}

Therefore, the required focal length is 280 mm280 \, \text{mm}.

Common mistakes

  • Using the inverse ratio f1f2\frac{f_1}{f_2} instead of f2f1\frac{f_2}{f_1} is incorrect because the output beam diameter increases with the output lens focal length. Use DoutDin=f2f1\frac{D_{out}}{D_{in}} = \frac{f_2}{f_1}.

  • Treating the problem as one of lens separation only is wrong. Although the lenses in a Keplerian beam expander are separated by f1+f2f_1 + f_2, the required beam diameter is determined by magnification, not by separation.

  • Substituting Din=14 mmD_{in} = 14 \, \text{mm} and Dout=2 mmD_{out} = 2 \, \text{mm} in reverse gives a shrinking system instead of an expanding system. Keep the input and output beam diameters in the correct order.

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