Animated Solution for Physics - Optics: An observer looks at a distance tree of height 10 m with a telescope of magnifying power of 20. To the observer the tree appears

Have you ever wondered how we actually perceive the size of an object? It all comes down to the visual angle.

Imagine you are standing in a vast field, looking at a distant tree. Light rays from the top and bottom of the tree travel to your eye, forming a small angle.

Let's call this angle $\alpha$. This angle dictates the size of the image formed on your retina, which your brain interprets as the height of the tree.

Now, you bring a telescope to your eye. What does the telescope actually do?

It doesn't physically move the tree, nor does it stretch the wood and leaves. Instead, it bends the light rays so that they enter your eye at a much steeper angle.

Let's call this new, larger angle $\beta$. The primary job of a telescope is to increase this visual angle, and we measure this ability using a term called magnifying power ($m$).

The magnifying power is defined as the ratio of the angle subtended by the image to the angle subtended by the object.

In our specific problem, we are given that the magnifying power of the telescope is $20$.

Let's substitute this value into our master equation.

By simply rearranging this equation, we can find the relationship between the new visual angle and the old one.

This is the magic moment! The angle $\beta$ is exactly $20$ times larger than $\alpha$.

Because the apparent height of an object is directly proportional to the visual angle it subtends at the eye, the tree now appears $20$ times taller.

Nearer: The Brain's Dilemma

Here is where many students make a classic mistake. You might be thinking, "If it looks bigger, shouldn't it appear $20$ times nearer?"

This is a very intuitive trap, but it is incorrect. Our perception of distance is incredibly complex and relies heavily on binocular vision (using two eyes) and accommodation (the flexing of our eye lenses).

When you look through a telescope, the instrument can be adjusted to form the virtual image at infinity. When an image is at infinity, your eye lenses are completely relaxed, just as they are when looking at the stars.

Because your eyes are not straining to focus on a close object, your brain does not receive the physiological cues that say "this object is near."

Instead, your brain simply registers a massive visual angle while your eyes remain relaxed. The result? The object doesn't feel closer; it just feels like a giant, towering version of itself.

Therefore, the most accurate and scientifically rigorous description is that the tree appears $20$ times taller.