Two plane mirrors are inclined at an angle of `60^(@)` with each other. An incident ray hits one of the mirror at an angle of `80^(@)` with the normal. Its angle of deviation after two reflections is

Options

A. `120^(@)`
B. `240^(@)`
C. zero
D. `60^(@)`

Answer

To solve the problem of finding the angle of deviation after two reflections from two plane mirrors inclined at an angle of \(60^\circ\), we can follow these steps:

### Step 1: Understand the setup
We have two plane mirrors inclined at an angle of \(60^\circ\). An incident ray strikes one of the mirrors at an angle of \(80^\circ\) with respect to the normal.

### Step 2: Determine the angle of incidence
The angle of incidence \(i_1\) with respect to the normal is given as \(80^\circ\). Therefore, the angle of reflection \(r_1\) will also be \(80^\circ\) (according to the law of reflection).

### Step 3: Calculate the angle of deviation after the first reflection
The angle of deviation \(D_1\) after the first reflection can be calculated using the formula:
\[ D_1 = i_1 - r_1 + \text{angle between mirrors} \]
Here, the angle between the mirrors is \(60^\circ\). Thus,
\[ D_1 = 80^\circ - 80^\circ + 60^\circ = 60^\circ \]

### Step 4: Determine the angle of incidence for the second mirror
After the first reflection, the ray will strike the second mirror. The angle of incidence \(i_2\) for the second mirror can be calculated as:
\[ i_2 = 180^\circ - (r_1 + \text{angle between mirrors}) = 180^\circ - (80^\circ + 60^\circ) = 40^\circ \]

### Step 5: Calculate the angle of reflection for the second mirror
According to the law of reflection, the angle of reflection \(r_2\) will also be \(40^\circ\).

### Step 6: Calculate the angle of deviation after the second reflection
The angle of deviation \(D_2\) after the second reflection can be calculated as:
\[ D_2 = i_2 - r_2 + \text{angle between mirrors} \]
Substituting the values:
\[ D_2 = 40^\circ - 40^\circ + 60^\circ = 60^\circ \]

### Step 7: Calculate the total angle of deviation
The total angle of deviation \(D\) after two reflections is the sum of the deviations from both reflections:
\[ D = D_1 + D_2 = 60^\circ + 60^\circ = 120^\circ \]

### Final Answer
The angle of deviation after two reflections is \(120^\circ\).

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