Animated Solution for Physics - Optics: A ray of light travelling in water is incident on its surface open to air. The angle of incidence is $\theta$, which is less than the critical angle. Then there will be

Reflection, Refraction, and the Angles Between

Imagine a ray of light traveling through water and striking the surface open to the air. It's not just a simple bounce; it's a split personality—part of it reflects, and part of it refracts. Since the angle of incidence, $\theta$, is less than the critical angle, total internal reflection doesn't happen. Both reflection and refraction occur simultaneously.

Let's look at the reflected ray first. According to the law of reflection, the angle it makes with the normal is exactly equal to the angle of incidence. So, this angle is also $\theta$.

Now, what about the refracted ray? Since light is moving from a denser medium (water) to a rarer medium (air), it speeds up and bends away from the normal. This means the angle of refraction, let's call it $r$, must be strictly greater than the angle of incidence, $\theta$. So, we have our first crucial inequality: $r>\theta$.

The question asks for the angle between the reflected and refracted rays. Let's call this angle $\alpha$. If you look at the straight line formed by the normal, the total angle from the bottom to the top is $180^{\circ }$. This total span is perfectly divided into three adjacent angles: the angle of reflection ($\theta$), the angle between the rays ($\alpha$), and the angle of refraction ($r$).

Therefore, we can write the geometric equation:

Rearranging our geometric equation to solve for $\alpha$ gives:

Here comes the final logical step. We established earlier that $r>\theta$. If we add $\theta$ to both sides of this inequality, we get:

Now, let's substitute this logic back into our equation for $\alpha$. Since we are subtracting a value ($\theta +r$) that is strictly larger than $2\theta$ from $180^{\circ }$, the resulting angle $\alpha$ must be strictly less than $180^{\circ }-2\theta$.

Mathematically:

And that perfectly matches option (c)! A simple geometric observation combined with Snell's Law leads to an elegant and rigorous inequality.