Understanding Critical Angle of Incidence

The question asks for the condition that defines the critical angle of incidence, denoted as $i_c$, when light travels from a medium where it is denser to a medium where it is rarer.

Light Bending from Denser to Rarer Medium

When a ray of light travels from a denser medium (higher refractive index, $n_1$) to a rarer medium (lower refractive index, $n_2$), it bends away from the normal. The angle of refraction ($\theta_r$) is greater than the angle of incidence ($i$).

Defining the Critical Angle

As the angle of incidence ($i$) increases, the angle of refraction ($\theta_r$) also increases. The critical angle of incidence ($i_c$) is a specific angle of incidence. It is defined as the angle of incidence for which the corresponding angle of refraction is exactly $90^\circ$. At this point, the refracted ray travels along the boundary surface between the two media.

• Angle of Incidence = $i_c$
• Angle of Refraction = $\theta_r = 90^\circ$

Applying Snell's Law

Snell's Law describes the relationship between the angles and refractive indices:

$ n_1 \sin(i) = n_2 \sin(\theta_r) $

At the critical angle ($i = i_c$), the angle of refraction is $90^\circ$ ($\theta_r = 90^\circ$). Substituting these values into Snell's Law:

$ n_1 \sin(i_c) = n_2 \sin(90^\circ) $

Since $\sin(90^\circ) = 1$, the equation becomes:

$ n_1 \sin(i_c) = n_2 $

This gives the formula for the critical angle:

$ \sin(i_c) = \frac{n_2}{n_1} $

This formula is valid only when $n_1 > n_2$ (denser to rarer medium), ensuring that $\frac{n_2}{n_1} < 1$, which is necessary for $\sin(i_c)$ to have a real value.

Analyzing the Options

• Option 1: the angle of reflection is $90^\circ$ - The angle of reflection is equal to the angle of incidence. This condition doesn't define the critical angle for refraction.
• Option 2: the angle of refraction is $90^\circ$ - This is the correct definition of the critical angle of incidence ($i_c$).
• Option 3: the angle of refraction is $0^\circ$ - This occurs when the angle of incidence is $0^\circ$ (the ray is perpendicular to the surface).
• Option 4: the angle of reflection is $0^\circ$ - This happens when the angle of incidence is $0^\circ$.

Therefore, the critical angle of incidence is the angle at which the angle of refraction becomes $90^\circ$.