Question 120: Ray Optics

Question

The refractive index of the material of a glass prism is $\sqrt{3}$. The angle of minimum deviation is equal to the angle of the prism. What is the angle of the prism?

Accepted Answer

$$ 	extbf{Step 1: Symmetry at Minimum Deviation} $$
In a prism, when the deviation is minimum, the path of light is symmetric. This means that the angle of incidence ($i$) is equal to the angle of emergence ($i'$), and the light inside the prism makes equal angles with the prism faces. If the prism angle is $$A$$, then the refracted angle at each interface is:
$$ r = \frac{A}{2} $$
$$ 	extbf{Step 2: Relating Deviation to the Angles} $$
The formula for the deviation ($D$) in a prism is given by:
$$ D = i + i' - A $$
At minimum deviation, $$i = i'$$, so:
$$ D_{	ext{min}} = 2i - A $$
The special condition given is that the minimum deviation is equal to the prism angle:
$$ D_{	ext{min}} = A $$
Thus:
$$ A = 2i - A \quad \Longrightarrow \quad 2i = 2A \quad \Longrightarrow \quad i = A $$
$$ 	extbf{Step 3: Applying Snell's Law} $$
At the first surface, Snell's law gives:
$$ \sin i = \mu \sin r $$
Substitute the values $$i = A$$ and $$r = \frac{A}{2}$$:
$$ \sin A = \mu \sin\frac{A}{2} $$
Given that the refractive index $$\mu = \sqrt{3}$$, we have:
$$ \sin A = \sqrt{3} \sin\frac{A}{2} $$
$$ 	extbf{Step 4: Solving the Equation} $$
Recall the double-angle formula for sine:
$$ \sin A = 2 \sin\frac{A}{2} \cos\frac{A}{2} $$
Substitute this into the previous equation:
$$ 2 \sin\frac{A}{2} \cos\frac{A}{2} = \sqrt{3} \sin\frac{A}{2} $$
Assuming $$\sin\frac{A}{2} 
eq 0$$, we can divide both sides by $$\sin\frac{A}{2}$$:
$$ 2 \cos\frac{A}{2} = \sqrt{3} $$
Solve for $$\cos\frac{A}{2}$$:
$$ \cos\frac{A}{2} = \frac{\sqrt{3}}{2} $$
Since:
$$ \cos 30^\circ = \frac{\sqrt{3}}{2} $$
It follows:
$$ \frac{A}{2} = 30^\circ \quad \Longrightarrow \quad A = 60^\circ $$
$$ 	extbf{Final Answer:} $$
The angle of the prism is $$60^\circ$$.

Answer

$60^{\circ}$

Answer

$50^{\circ}$

Answer

$58^{\circ}$

Answer

$48^{\circ}$

Ray Optics

The refractive index of the material of a glass prism is \sqrt{3}. The angle of minimum deviation is equal to the angle of the prism. What is the angle of the prism?

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