The index of refraction of heavy flint glass is `1.68` at `434 nm` and `1.65` at `671nm`. Calculate the difference in the angle of deviation of blue `(434 nm)` and red `(671 nm)` light incident at `65^@` on one side of a heavy flint glass prism with apex angle `60^@`.

Explanation

To solve the problem of calculating the difference in the angle of deviation of blue (434 nm) and red (671 nm) light incident at 65° on a heavy flint glass prism with an apex angle of 60°, we will follow these steps:

### Step 1: Calculate the angle of refraction for blue light (R1)
Using Snell's Law:
\[ \mu = \frac{\sin I}{\sin R} \]
Where:
- \( \mu \) is the refractive index of the glass for blue light (1.68),
- \( I \) is the angle of incidence (65°),
- \( R1 \) is the angle of refraction for blue light.

Rearranging gives:
\[ \sin R1 = \frac{\sin I}{\mu} \]
Substituting the values:
\[ \sin R1 = \frac{\sin 65°}{1.68} \]
Calculating \( R1 \):
\[ R1 \approx 32.6° \]

### Step 2: Calculate the angle of refraction for red light (R1)
Using the same approach for red light with a refractive index of 1.65:
\[ \sin R1 = \frac{\sin 65°}{1.65} \]
Calculating \( R1 \):
\[ R1 \approx 33.3° \]

### Step 3: Calculate the second angle of refraction (R2) for both colors
Using the formula for the angles in a prism:
\[ R2 = A - R1 \]
Where \( A \) is the apex angle of the prism (60°).

For blue light:
\[ R2_{blue} = 60° - 32.6° = 27.4° \]

For red light:
\[ R2_{red} = 60° - 33.3° = 26.7° \]

### Step 4: Calculate the angle of incidence at the second face (I2)
Using Snell's Law again:
\[ \sin I2 = \mu \cdot \sin R2 \]

For blue light:
\[ \sin I2_{blue} = 1.68 \cdot \sin 27.4° \]
Calculating \( I2_{blue} \):
\[ I2_{blue} \approx 50.6° \]

For red light:
\[ \sin I2_{red} = 1.65 \cdot \sin 26.7° \]
Calculating \( I2_{red} \):
\[ I2_{red} \approx 47.8° \]

### Step 5: Calculate the angle of deviation for both colors
The angle of deviation \( D \) is given by:
\[ D = I1 + I2 - A \]

For blue light:
\[ D_{blue} = 65° + 50.6° - 60° = 55.6° \]

For red light:
\[ D_{red} = 65° + 47.8° - 60° = 52.8° \]

### Step 6: Calculate the difference in the angle of deviation
\[ \Delta D = D_{blue} - D_{red} \]
Calculating the difference:
\[ \Delta D = 55.6° - 52.8° = 2.8° \]

Thus, the difference in the angle of deviation of blue and red light is **2.8°**.

---