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The refracting angle of a prism is $\mathrm{A}$ and refractive index of the material of the prism is $\cot (\mathrm{A} / 2)$ . Then the angle of minimum deviation will be -

Option: 1
$180-2 \mathrm{~A}$

Option: 2
$90-\mathrm{A}$

Option: 3
$180+2 \mathrm{~A}$

Option: 4
$180-3 \mathrm{A}$

Answers (1)

For Prism

$\mathrm{\mu}=\frac{\sin \left(\frac{A+\delta m}{2}\right)}{\sin (A / 2)}$

$\mu \rightarrow \text{refractive index}$

$\text{A} \rightarrow\text{refracting angle of prism}$

$\mathrm{\delta_m} \rightarrow\text{minimum deviation angle }$

$\mathrm{\mu}=\cot \frac{A}{2} \\$

$\mathrm{\cot \left(\frac{A}{2}\right)=\frac{\sin \left(\frac{A+\delta _{m}}{2}\right)}{\sin (A / 2)}} \\$

$\mathrm{\frac{\cos \left(\frac{A}{2}\right)}{\sin (A / 2)}=\frac{\sin \left(A+\frac{\delta_{m}}{2}\right)}{\sin (A / 2)} }$

$\mathrm{\cos \left(\frac{A}{2}\right)=\sin \left(\frac{A+\delta _{m}}{2}\right)} \\$

$\mathrm{90^{\circ}-\frac{A}{2}=\frac{A+\delta _{m}}{2}} \\$

$\mathrm{90^{\circ}-A=\frac{\delta _{m}}{2}} \\$

$\mathrm{\delta_{m}=180^{\circ}-2 A }$

Hence the correct option is 1

Posted by

Sumit Saini

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