Animated Solution for Physics - Optics: Wavelength of light used in an optical instrument are ${\lambda }_1=4000\text{ A˚}$ and ${\lambda }_2=5000\text{ A˚}$, then ratio of their respective resolving powers (corresponding to ${\lambda }_1$ and ${\lambda }_2$) is

Understanding Resolving Power

Imagine you are looking up at the night sky, trying to spot a distant binary star system. To the naked eye, they might look like a single, blurry point of light. But when you look through a powerful telescope, suddenly, that single point splits into two distinct, brilliant stars. This magical ability of an optical instrument to distinguish between two closely spaced objects is known as its Resolving Power.

In the world of optics, whether you are peering through a microscope at a tiny cell or gazing through a telescope at a distant galaxy, resolving power is your best friend. But what determines this power? It turns out, the secret lies in the very nature of the light you are using.

Wavelength and Resolution

When light passes through the circular aperture of an optical instrument, it diffracts, creating a pattern of concentric rings known as an Airy disk. According to Rayleigh's criterion, two point sources are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other.

Mathematically, the limit of resolution ($\Delta \theta$ for telescopes or $\Delta d$ for microscopes) is directly proportional to the wavelength of the light ($\lambda$). Because Resolving Power (RP) is the reciprocal of this limit of resolution, we arrive at a beautiful and simple inverse relationship:

This means that the smaller the wavelength of the light, the higher the resolving power. Blue light, with its shorter wavelength, will always give you a sharper, more resolved image than red light!

In our specific problem, we are given two different wavelengths of light being used in the same optical instrument: - ${\lambda }_1=4000\text{ A˚}$ (which is in the violet/blue region) - ${\lambda }_2=5000\text{ A˚}$ (which is in the green/yellow region)

We need to find the ratio of their respective resolving powers, ${\text{RP}}_1$ to ${\text{RP}}_2$. Using our inverse proportionality rule, we can set up the following equation:

Notice how the subscripts flip on the right side of the equation. This is the direct mathematical consequence of the inverse relationship.

Now, all that is left is to substitute our given values into the equation.

Here is a pro-tip: because we are calculating a ratio, the units of Angstroms ($\text{A˚}$) perfectly cancel each other out. There is absolutely no need to waste time converting these values into meters!

Dividing both the numerator and the denominator by $1000$, we get:

And there we have it! The ratio of their respective resolving powers is $5:4$. This confirms our earlier intuition: the shorter wavelength (${\lambda }_1=4000\text{ A˚}$) yields a higher resolving power compared to the longer wavelength.

While wavelength plays a crucial role, it is not the only factor. The resolving power is also directly proportional to the size of the aperture ($a$) of the instrument. This is exactly why astronomers build massive telescopes with mirrors that are several meters across—to capture more light and achieve an incredibly high resolving power, allowing us to peer deeper and clearer into the cosmos!