Question

The angle of minimum deviation for a 90° prism is 30°. What is the speed of light in the prism?

• A. 2.4×108 ms-1
• B. 2.1×108 ms-1
• C. 1.8×108 ms-1
• D. 2.2×108 ms-1

Answer 1

Answer: (A)

2.4×108 ms-1

Answer 2

To find the speed of light in the prism, we first need to determine the refractive index of the prism material.

Given:

1. Angle of the prism, $A = 90^\circ$
2. Angle of minimum deviation, $\delta_m = 30^\circ$
3. Speed of light in vacuum, $c = 3 \times 10^8 \text{ m/s}$

The formula for the refractive index ($n$) of a prism in terms of the prism angle ($A$) and the angle of minimum deviation ($\delta_m$) is: $n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$

Substitute the given values: $\frac{A + \delta_m}{2} = \frac{90^\circ + 30^\circ}{2} = \frac{120^\circ}{2} = 60^\circ$ $\frac{A}{2} = \frac{90^\circ}{2} = 45^\circ$

Now, substitute these into the refractive index formula: $n = \frac{\sin(60^\circ)}{\sin(45^\circ)}$ We know that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$ and $\sin(45^\circ) = \frac{1}{\sqrt{2}}$. $n = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{\sqrt{2}}} = \frac{\sqrt{3}}{2} \times \sqrt{2} = \frac{\sqrt{6}}{2}$

Next, we use the relationship between the refractive index ($n$), the speed of light in vacuum ($c$), and the speed of light in the prism ($v$): $n = \frac{c}{v}$

Rearrange the formula to solve for $v$: $v = \frac{c}{n}$

Substitute the value of $c$ and the calculated value of $n$: $v = \frac{3 \times 10^8 \text{ m/s}}{\frac{\sqrt{6}}{2}}$ $v = \frac{2 \times 3 \times 10^8}{\sqrt{6}} \text{ m/s}$ $v = \frac{6 \times 10^8}{\sqrt{6}} \text{ m/s}$

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{6}$: $v = \frac{6 \sqrt{6}}{6} \times 10^8 \text{ m/s}$ $v = \sqrt{6} \times 10^8 \text{ m/s}$

Now, calculate the numerical value of $\sqrt{6}$: $\sqrt{6} \approx 2.449$

So, $v \approx 2.449 \times 10^8 \text{ m/s}$. Rounding to one decimal place, $v \approx 2.4 \times 10^8 \text{ m/s}$.

Comparing this with the given options, the closest value is $2.4 \times 10^8 \text{ ms}^{-1}$.