A flint glass prism and a crown glass prism are to be combined in such a way that the deviation of the mean ray is zero. The refractive index of flint and crown glasses for the mean ray are 1.620 and 1.518 respectively. If the refracting angle of the flint prism is `6.0^@`, what would be the refracting angle of the crown prism ?

Explanation

To solve the problem of finding the refracting angle of the crown glass prism such that the net deviation of the mean ray is zero, we can follow these steps:

### Step 1: Understand the Concept of Deviation
The deviation of light passing through a prism can be expressed using the formula:
\[ \Delta = (\mu - 1)A \]
where \(\Delta\) is the angle of deviation, \(\mu\) is the refractive index of the prism, and \(A\) is the refracting angle of the prism.

### Step 2: Set Up the Equations
For the flint glass prism:
\[ \Delta_f = (\mu_f - 1)A_f \]
For the crown glass prism:
\[ \Delta_c = (\mu_c - 1)A_c \]
Given that the net deviation is zero:
\[ \Delta_f = \Delta_c \]

### Step 3: Substitute the Given Values
From the problem, we have:
- Refractive index of flint glass, \(\mu_f = 1.620\)
- Refractive index of crown glass, \(\mu_c = 1.518\)
- Refracting angle of the flint prism, \(A_f = 6.0^\circ\)

Substituting these values into the equations:
\[ (\mu_f - 1)A_f = (\mu_c - 1)A_c \]
\[ (1.620 - 1) \cdot 6.0 = (1.518 - 1) \cdot A_c \]

### Step 4: Simplify the Equation
Calculating the left side:
\[ 0.620 \cdot 6.0 = 3.72 \]
Calculating the right side:
\[ 0.518 \cdot A_c \]
Thus, we have:
\[ 3.72 = 0.518 \cdot A_c \]

### Step 5: Solve for \(A_c\)
To find \(A_c\):
\[ A_c = \frac{3.72}{0.518} \]
Calculating this gives:
\[ A_c \approx 7.17^\circ \]

### Step 6: Round the Answer
Rounding to one decimal place, we find:
\[ A_c \approx 7.2^\circ \]

### Final Answer
The refracting angle of the crown glass prism is approximately \(7.2^\circ\).

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