In the given figure, the face $$AC$$ of the equilateral prism is immersed in a liquid of refractive index $$n$$. For incident angle $$60°$$ at the side $$AC$$, the refracted light beam just grazes along face $$AC$$. The refractive index of the liquid $$n = \dfrac{\sqrt{x}}{4}$$. The value of $$x$$ is ______. (Given refractive index of glass $$= 1.5$$)

We need to determine the value of $$x$$ given that the refracted light beam just grazes along the face $$AC$$ of an equilateral glass prism immersed in a liquid of refractive index $$n = \frac{\sqrt{x}}{4}$$.

1. Identify the Angles at Face $$AB$$ and Face $$AC$$

From the diagram 

• The prism is equilateral, so the angle of the prism is $$A = 60^\circ$$.
• The light ray enters normally at the first surface $$AB$$ because it enters perpendicular to that surface. Thus, the angle of refraction at the first surface is $$r_1 = 0^\circ$$.
• The relationship between the prism angle and the internal angles is given by $$A = r_1 + r_2$$. Substituting our values gives the angle of incidence at the second face $$AC$$:

  $$r_2 = A - r_1 = 60^\circ - 0^\circ = 60^\circ$$

• At the face $$AC$$, the refracted light beam just grazes along the surface, which means the angle of refraction in the liquid is $$r = 90^\circ$$. This implies $$60^\circ$$ is the critical angle for the glass-liquid interface.

2. Apply Snell's Law at Face $$AC$$ and Solve for $$x$$

Applying Snell's Law at the interface between the glass ($\mu_{\text{glass}} = 1.5 = \frac{3}{2}$) and the surrounding liquid ($$n$$):

$$\mu_{\text{glass}} \cdot \sin(r_2) = n \cdot \sin(r)$$

Substitute the known angles and values into the equation:

$$\frac{3}{2} \cdot \sin(60^\circ) = n \cdot \sin(90^\circ)$$

$$\frac{3}{2} \cdot \frac{\sqrt{3}}{2} = n \cdot 1 \implies n = \frac{3\sqrt{3}}{4}$$

Now, equate this result to the expression provided in the problem statement ($$n = \frac{\sqrt{x}}{4}$$):

$$\frac{\sqrt{x}}{4} = \frac{3\sqrt{3}}{4}$$

$$\sqrt{x} = 3\sqrt{3}$$

Squaring both sides of the equation to isolate $$x$$:

$$x = (3\sqrt{3})^2 = 9 \times 3 = 27$$

Conclusion

The value of $$x$$ is 27.