To find the minimum deviation angle [tex]\(\delta_m\)[/tex] produced by a prism with an angle [tex]\(A = 60^\circ\)[/tex] and a refractive index [tex]\(\mu = 2\)[/tex], we can use the formula for the angle of minimum deviation for a prism:
[tex]\[
\delta_m = 2 \arcsin\left(\mu \cdot \sin\left(\frac{A}{2}\right)\right) - A
\][/tex]
Let's solve this step-by-step:
1. Convert the prism angle from degrees to radians:
[tex]\[
A = 60^\circ
\][/tex]
We need this angle in radians:
[tex]\[
A_{\text{rad}} = \frac{60 \times \pi}{180} = \frac{\pi}{3} \approx 1.0471975511965976 \, \text{radians}
\][/tex]
2. Calculate the half-angle of the prism in radians:
[tex]\[
\frac{A}{2} = \frac{60^\circ}{2} = 30^\circ
\][/tex]
In radians:
[tex]\[
\left(\frac{A}{2}\right)_{\text{rad}} = \frac{30 \times \pi}{180} = \frac{\pi}{6} \approx 0.5235987755982988 \, \text{radians}
\][/tex]
3. Calculate [tex]\(\sin\left(\frac{A}{2}\right)\)[/tex]:
[tex]\[
\sin\left(\frac{A}{2}\right) = \sin\left(30^\circ\right) = \frac{1}{2} = 0.5
\][/tex]
4. Calculate the term inside the arcsine function:
[tex]\[
\mu \cdot \sin\left(\frac{A}{2}\right) = 2 \cdot 0.5 = 1
\][/tex]
However, due to numerical precision, this value might slightly differ, for example:
[tex]\[
\mu \cdot \sin\left(\frac{A}{2}\right) \approx 0.9999999999999999
\][/tex]
5. Calculate the arcsine of this term:
[tex]\[
\arcsin\left(0.9999999999999999\right) \approx \arcsin(1) = \frac{\pi}{2} \approx 1.5707963118937354 \, \text{radians}
\][/tex]
6. Calculate the minimum deviation in radians:
[tex]\[
\delta_{m, \text{rad}} = 2 \cdot \arcsin\left(\mu \cdot \sin\left(\frac{A}{2}\right)\right) - A_{\text{rad}}
\][/tex]
Substitute the values we have calculated:
[tex]\[
\delta_{m, \text{rad}} = 2 \times 1.5707963118937354 - 1.0471975511965976 \approx 2.0943950725908733 \, \text{radians}
\][/tex]
7. Convert the minimum deviation from radians back to degrees:
[tex]\[
\delta_{m} \approx 2.0943950725908733 \times \frac{180}{\pi} \approx 119.99999829245273^\circ
\][/tex]
Thus, the minimum deviation ([tex]\(\delta_m\)[/tex]) for yellow light in this prism is approximately:
[tex]\[
\delta_m \approx 119.99999829245273^\circ
\][/tex]
