A thin prism of crown glass `(mu_r=1.515, mu_v=1.525) ` and a thin prism of flint glass `(mu_r=1.612, mu_v=1.632)` are placed in contact with each other. Their refracting angles are `5.0^@` each and are similarly directed. Calculate the angular dispersion produced by the combination.

Answer

To solve the problem of calculating the angular dispersion produced by the combination of a thin prism of crown glass and a thin prism of flint glass, we can follow these steps:

### Step 1: Understand the Problem
We have two prisms:
- Crown glass with refractive indices:
- \( \mu_r = 1.515 \) (for red light)
- \( \mu_v = 1.525 \) (for violet light)
- Flint glass with refractive indices:
- \( \mu_r = 1.612 \) (for red light)
- \( \mu_v = 1.632 \) (for violet light)

Both prisms have a refracting angle \( A = 5^\circ \).

### Step 2: Calculate the Deviation for Each Prism
The deviation \( \delta \) produced by a prism can be calculated using the formula:
\[ \delta = (\mu - 1) \cdot A \]
where \( \mu \) is the refractive index of the prism for the specific color of light, and \( A \) is the angle of the prism.

### Step 3: Calculate the Deviation for Violet Light
For the crown glass (violet light):
\[ \delta_C = (\mu_{vC} - 1) \cdot A = (1.525 - 1) \cdot 5^\circ = 0.525 \cdot 5^\circ = 2.625^\circ \]

For the flint glass (violet light):
\[ \delta_F = (\mu_{vF} - 1) \cdot A = (1.632 - 1) \cdot 5^\circ = 0.632 \cdot 5^\circ = 3.16^\circ \]

### Step 4: Calculate the Total Deviation for Violet Light
The total deviation for violet light when both prisms are placed in contact:
\[ \delta_V = \delta_C + \delta_F = 2.625^\circ + 3.16^\circ = 5.785^\circ \]

### Step 5: Calculate the Deviation for Red Light
For the crown glass (red light):
\[ \delta_C = (\mu_{rC} - 1) \cdot A = (1.515 - 1) \cdot 5^\circ = 0.515 \cdot 5^\circ = 2.575^\circ \]

For the flint glass (red light):
\[ \delta_F = (\mu_{rF} - 1) \cdot A = (1.612 - 1) \cdot 5^\circ = 0.612 \cdot 5^\circ = 3.06^\circ \]

### Step 6: Calculate the Total Deviation for Red Light
The total deviation for red light:
\[ \delta_R = \delta_C + \delta_F = 2.575^\circ + 3.06^\circ = 5.635^\circ \]

### Step 7: Calculate the Angular Dispersion
The angular dispersion \( D \) is given by:
\[ D = \delta_V - \delta_R \]
Substituting the values we calculated:
\[ D = 5.785^\circ - 5.635^\circ = 0.15^\circ \]

### Final Answer
The angular dispersion produced by the combination of the two prisms is \( 0.15^\circ \).

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