For total internal reflection to take place, the angle of incidence `i` and the refractive index `mu` of the medium must satisfy the inequality

Options

A. `mu = sin c`
B. `mu = tan c`
C. `(1/mu) = sin c`
D. `(1/mu)= cos c`

Answer

To determine the conditions for total internal reflection to occur, we need to analyze the relationship between the angle of incidence (i) and the refractive index (µ) of the medium. Here’s a step-by-step solution:

### Step 1: Understand Total Internal Reflection
Total internal reflection occurs when light travels from a denser medium to a rarer medium and the angle of incidence exceeds a certain critical angle (C).

### Step 2: Define Critical Angle
The critical angle (C) is defined as the angle of incidence in the denser medium at which the angle of refraction in the rarer medium is 90 degrees.

### Step 3: Apply Snell's Law
Using Snell's Law, which states:
\[ \mu \sin(i) = \mu_1 \sin(r) \]
where:
- \( \mu \) is the refractive index of the denser medium,
- \( \mu_1 \) is the refractive index of the rarer medium (for air, it is approximately 1),
- \( i \) is the angle of incidence,
- \( r \) is the angle of refraction.

### Step 4: Set Up the Equation for Critical Angle
At the critical angle (C), the angle of refraction (r) is 90 degrees. Therefore, we can substitute \( r = 90^\circ \) into Snell's Law:
\[ \mu \sin(C) = \mu_1 \sin(90^\circ) \]
Since \( \sin(90^\circ) = 1 \), this simplifies to:
\[ \mu \sin(C) = \mu_1 \]

### Step 5: Solve for the Critical Angle
Rearranging the equation gives us:
\[ \sin(C) = \frac{\mu_1}{\mu} \]
For air, where \( \mu_1 = 1 \):
\[ \sin(C) = \frac{1}{\mu} \]

### Step 6: Establish the Condition for Total Internal Reflection
For total internal reflection to occur, the angle of incidence (i) must be greater than the critical angle (C):
\[ i > C \]
This means:
\[ i > \sin^{-1}\left(\frac{1}{\mu}\right) \]

### Final Conclusion
Thus, the inequality that must be satisfied for total internal reflection to take place is:
\[ i > \sin^{-1}\left(\frac{1}{\mu}\right) \]