Animated Solution for Physics - Optics: The focal length of a thin biconvex lens is $20\text{ cm}$. When an object is moved from a distance of $25\text{ cm}$ in front of it to $50\text{ cm}$, the magnification of its image changes from $m_{25}$ to $m_{50}$. The ratio $\frac{m_{25}}{m_{50}}$ is

The problem asks us to find the ratio of magnifications of an object placed at two different positions in front of a biconvex lens.

We are given two scenarios. In the first scenario, the object is placed at a distance of $25\text{ cm}$ in front of the lens. In the second scenario, it is moved further away to a distance of $50\text{ cm}$.

We need to find the magnification in both cases, denoted as $m_{25}$ and $m_{50}$, and then calculate their ratio.

To find the magnification, we could use the standard lens formula $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$ to find the image distance $v$ for each case, and then use the magnification formula $m=\frac{v}{u}$.

However, there is a much more elegant way! Let's derive a direct formula for magnification in terms of $u$ and $f$.

Starting with the lens formula:

Multiply the entire equation by $u$:

Rearranging the terms, we get:

Since magnification $m=\frac{v}{u}$, we can just take the reciprocal:

This is a powerful shortcut that saves us from calculating $v$ explicitly!

Now, let's apply our master equation to both scenarios. Watch out for the sign convention! Since the object is placed in front of the lens, the object distance $u$ will be negative.

Case 1: Object at $25\text{ cm}$ Here, $u_1=-25\text{ cm}$. Substituting this into our formula: The magnification is $-4$, which means the image is real, inverted, and four times larger than the object.

Case 2: Object at $50\text{ cm}$ Here, $u_2=-50\text{ cm}$. Substituting this into our formula: The magnification is $-\frac{2}{3}$, meaning the image is still real and inverted, but now it is diminished (smaller than the object).

Let's plug in the values we just calculated:

The ratio of the magnifications is exactly $6$.

Notice how moving the object away from the focus drastically reduced the magnification. This is a fundamental property of convex lenses: as the object moves from the focus towards infinity, the real image moves from infinity towards the focus and becomes progressively smaller!