A ray of light is incident at an angle of 60$$^\circ$$ on one face of a prism of angle 30$$^\circ$$. The emergent ray of light makes an angle of 30$$^\circ$$ with incident ray. The angle made by the emergent ray with second face of prism will be:

We have a prism whose refracting angle is given as $$A = 30^\circ$$. A light ray falls on the first face of the prism making an external angle of incidence $$i = 60^\circ$$ with the normal to that face.

After refraction through the prism the ray emerges from the second face. The angle between the original incident ray and the finally emergent ray is called the angle of deviation. It is given in the statement that this angle equals $$30^\circ$$; therefore we may write $$\delta = 30^\circ$$.

For any prism the standard relation connecting the angle of deviation $$\delta$$, the angle of the prism $$A$$, the external angle of incidence $$i$$ and the external angle of emergence $$e$$ is first stated:

$$\boxed{\;\delta = i + e - A\;}$$

Now we substitute the known numerical values. Putting $$\delta = 30^\circ$$, $$i = 60^\circ$$ and $$A = 30^\circ$$ in the formula we get

$$30^\circ = 60^\circ + e - 30^\circ.$$

Combining the terms on the right-hand side gives

$$30^\circ = 30^\circ + e.$$

Shifting $$30^\circ$$ to the left side, we obtain

$$e = 30^\circ - 30^\circ = 0^\circ.$$

This result means that the emergent ray is along the normal to the second face, because the external angle that the ray makes with that face’s normal is zero.

We are, however, asked for the angle that the emergent ray makes with the plane (surface) of the second face. A ray that is normal to a surface is perpendicular to that surface. The angle between a line and a plane that are perpendicular is $$90^\circ$$.

So, the emergent ray makes an angle of $$90^\circ$$ with the second face of the prism.

Hence, the correct answer is Option B.