Question

A thin convex lens of focal length \( 5 \) cm and a thin concave lens of focal length \( 4 \) cm are combined together (without any gap), and this combination has magnification \( m_1 \) when an object is placed \( 10 \) cm before the convex lens.

Keeping the positions of the convex lens and the object undisturbed, a gap of \( 1 \) cm is introduced between the lenses by moving the concave lens away. This leads to a change in magnification of the total lens system to \( m_2 \).

The value of \( \dfrac{m_1}{m_2} \) is

• A. $\dfrac{25}{27}$
• B. $\dfrac{3}{2}$
• C. $\dfrac{5}{27}$
• D. $\dfrac{5}{9}$

Hint:

When lenses are separated, treat image of the first lens as the object for the second lens and calculate magnifications stepwise.

Answer 1

The Correct Option is A

Step 1: Find image formed by the convex lens.
For the convex lens: \[ f_1 = +5\text{ cm}, \quad u_1 = -10\text{ cm} \] Using lens formula: \[ \frac{1}{f_1} = \frac{1}{v_1} - \frac{1}{u_1} \Rightarrow \frac{1}{5} = \frac{1}{v_1} + \frac{1}{10} \] \[ \frac{1}{v_1} = \frac{1}{10} \Rightarrow v_1 = 10\text{ cm} \] Magnification due to convex lens: \[ m_1^{(1)} = \frac{v_1}{u_1} = -1 \] Step 2: Case I (Lenses in contact).
Equivalent focal length: \[ \frac{1}{f} = \frac{1}{5} - \frac{1}{4} = -\frac{1}{20} \Rightarrow f = -20\text{ cm} \] Using lens formula: \[ \frac{1}{-20} = \frac{1}{v} - \frac{1}{-10} \Rightarrow \frac{1}{v} = -\frac{3}{20} \Rightarrow v = -\frac{20}{3} \] Total magnification: \[ m_1 = \frac{v}{u} = \frac{-20/3}{-10} = \frac{2}{3} \] Step 3: Case II (Gap of 1 cm introduced).
Image formed by convex lens is $10$ cm to the right.
Concave lens is $1$ cm away, so object distance for concave lens: \[ u_2 = -9\text{ cm}, \quad f_2 = -4\text{ cm} \] Using lens formula: \[ \frac{1}{-4} = \frac{1}{v_2} - \frac{1}{-9} \Rightarrow \frac{1}{v_2} = -\frac{13}{36} \Rightarrow v_2 = -\frac{36}{13} \] Magnification due to concave lens: \[ m_2^{(2)} = \frac{v_2}{u_2} = \frac{4}{13} \] Total magnification: \[ m_2 = (-1)\times\frac{4}{13} = \frac{4}{13} \] Step 4: Compute the ratio.
\[ \frac{m_1}{m_2} = \frac{\frac{2}{3}}{\frac{4}{13}} = \frac{25}{27} \] Final Answer: $\boxed{\dfrac{25}{27}}$