Question

Monochromatic light is incident on a glass prism of angle A. If the refractive index of the material oft the prism is μ, a ray incident at an angle θ, on the face AB would get transmitted through the face AC of the prism provided

$straight theta space greater than space sin to the power of negative 1 end exponent open square brackets straight mu space sin space open parentheses straight A minus sin to the power of negative 1 end exponent open parentheses 1 over straight mu close parentheses close parentheses close square brackets$
$straight theta space less than space sin to the power of negative 1 end exponent open square brackets straight mu space sin space open parentheses straight A minus sin to the power of negative 1 end exponent open parentheses 1 over straight mu close parentheses close parentheses close square brackets$
$straight theta less than cos to the power of negative 1 end exponent open square brackets straight mu space sin space open parentheses straight A minus sin to the power of negative 1 end exponent open parentheses 1 over straight mu close parentheses close parentheses close square brackets$

$straight theta space greater than cos to the power of negative 1 end exponent open square brackets straight mu space sin space open parentheses straight A minus sin to the power of negative 1 end exponent open parentheses 1 over straight mu close parentheses close parentheses close square brackets$

Solution

straight theta space greater than space sin to the power of negative 1 end exponent open square brackets straight mu space sin space open parentheses straight A minus sin to the power of negative 1 end exponent open parentheses 1 over straight mu close parentheses close parentheses close square brackets

The Ray will get transmitted through face AC if i_AC < i_C
Consider the ray diagram is shown below θ.

A ray of light incident on face AB at angle
r₁ = Angle of refraction on face AB
r₂ = Angle of incidence at face AC
For transmission of light through face AC
i_AC < i_C or A-r_i < i_C
or Sin (A-r₁)1)< 1/μ
$straight A minus straight r subscript 1 space less than space sin to the power of negative 1 end exponent space open parentheses 1 over straight mu close parentheses sin space straight r subscript 1 space greater than space sin space open square brackets straight A minus sin to the power of negative 1 end exponent open parentheses 1 over straight mu close parentheses close square brackets$
Now, applying snell's law at the face AB
1 x sin θ = μ sin r1 or sinr1 = sinθ/μ
$fraction numerator sin space straight theta over denominator straight mu end fraction space greater than space sin space open square brackets straight A minus sin to the power of negative 1 end exponent open parentheses 1 over straight mu close parentheses close square brackets straight theta greater than space sin to the power of negative 1 end exponent open square brackets straight mu space sin space open curly brackets straight A space minus space sin to the power of negative 1 end exponent open parentheses 1 over straight mu close parentheses close curly brackets close square brackets$

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