Ratio of Magnifications for Biconvex Lens

To find the ratio of magnifications $\frac{\mathrm{m}_{25}}{\mathrm{m}_{50}}$, we use the lens formula and the definition of magnification for a thin lens.

1. Lens Formula and Magnification Formula

According to the thin lens formula:
$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

where:

• $v$ is the image distance,
• $u$ is the object distance,
• $f$ is the focal length of the lens.

Multiplying the entire equation by $u$:
$\frac{u}{v} - 1 = \frac{u}{f} \implies \frac{u}{v} = \frac{u + f}{f}$

Since magnification $m = \frac{v}{u}$, we get:
$m = \frac{f}{f + u}$

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2. Calculation of Magnifications

For a biconvex (converging) lens, the focal length is positive:
$f = +20\text{ cm}$

Using the standard sign convention, the object placed in front of the lens has a negative object distance:

• For object distance $u_1 = -25\text{ cm}$:
  $m_{25} = \frac{20}{20 + (-25)} = \frac{20}{-5} = -4$

• For object distance $u_2 = -50\text{ cm}$:
  $m_{50} = \frac{20}{20 + (-50)} = \frac{20}{-30} = -\frac{2}{3}$

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3. Ratio of Magnifications

The ratio $\frac{\mathrm{m}_{25}}{\mathrm{m}_{50}}$ is calculated as:
$\frac{\mathrm{m}_{25}}{\mathrm{m}_{50}} = \frac{-4}{-\frac{2}{3}} = \frac{4 \times 3}{2} = 6$

Thus, the ratio $\frac{\mathrm{m}_{25}}{\mathrm{m}_{50}}$ is 6.