A glass block whose refractive index is 1.564 for sodium light is to be used to construct a prism such that the angle of minimum deviation for such light shall be equal to the angle of the prism.what is the angle of the prism?

Answer 1

To find the angle of the prism, we can use the formula for the angle of minimum deviation for a prism:

\[ \theta_m = \frac{A + \delta_m}{2} \]

Where:
- \( \theta_m \) is the angle of minimum deviation
- \( A \) is the angle of the prism
- \( \delta_m \) is the minimum deviation angle for the prism

The minimum deviation angle for a prism is given by the formula:

\[ \delta_m = A - \frac{A}{n} \]

Where:
- \( n \) is the refractive index of the prism material

Given that the refractive index of the glass block is 1.564, we can substitute this value into the formula for the minimum deviation angle and then substitute it into the formula for the angle of the prism:

\[ \delta_m = A - \frac{A}{1.564} \]
\[ \delta_m = A(1 - \frac{1}{1.564}) \]
\[ \delta_m = A \times 0.3615 \]
\[ \delta_m = 0.3615A \]

Now we substitute this into the formula for the angle of minimum deviation:

\[ \theta_m = \frac{A + 0.3615A}{2} \]
\[ \theta_m = \frac{1.3615A}{2} \]
\[ \theta_m = 0.68075A \]

Since we are given that the angle of minimum deviation is equal to the angle of the prism, we can set these two angles equal to each other:

\[ A = 0.68075A \]
\[ A = A \times 0.68075 \]
\[ 1 = 0.68075 \]

Therefore, the angle of the prism is \[ 1 \times \frac{180}{\pi} \approx 54.7^\circ \].