Answer
(a) angle of refraction $\theta_{2}=45.78^{0}$
(b) angle of refraction $\theta_{2}=50.17^{0}$, drawing incorrect.
(c ) angle of refraction$\theta_{2}=69.41^{0}$
(b) angle of refraction $\theta_{2}=0^{0}$, drawing incorrect.
Work Step by Step
When light travels from medium $1$ to medium $2$
according to Snell's law
$n_{1}sin\theta_{1}=n_{2}sin\theta_{2}$
Here
$n_{1}$ is refractive index of medium $1$
$\theta_{1} $ is angle of incidence in medium $1$
$n_{2}$ is refractive index of medium $2$
$\theta_{2} $ is angle of refraction in medium $2$
we can rewrite above equation as
$sin\theta_{2}=\frac{n_{1}}{n_{2}}\times sin\theta_{1}$
or angle of refraction $\theta_{2}=sin^{-1}(\frac{n_{1}}{n_{2}}\times sin\theta_{1})$
case (a) $n_{1}=1.4$ and $n_{2}=1.6$
angle of incidence $\theta_{1}=55^{0}$
$\theta_{2}=sin^{-1}(\frac{n_{1}}{n_{2}}\times sin\theta_{1})=sin^{-1}
(\frac{1.4}{1.6}\times sin55^{0})$
$\theta_{2}=sin^{-1}(0.875\times 0.81915)=sin^{-1}(0.71675)$
$\theta_{2}=45.78^{0}$
refraction shown by the drawing is correct.
case (b) $n_{1}=1.5$ and $n_{2}=1.6$
angle of incidence $\theta_{1}=55^{0}$
$\theta_{2}=sin^{-1}(\frac{n_{1}}{n_{2}}\times sin\theta_{1})=sin^{-1}
(\frac{1.5}{1.6}\times sin55^{0})$
$\theta_{2}=sin^{-1}(0.9375\times 0.81915)=sin^{-1}(0.76795)$
$\theta_{2}=50.17^{0}$
refraction shown by the drawing is incorrect. Because in drawing angle of refraction is looking grater than angle of incidence , whereas from calculation angle of refraction is smaller than angle of incidence.
case (c) $n_{1}=1.6$ and $n_{2}=1.4$
angle of incidence $\theta_{1}=55^{0}$
$\theta_{2}=sin^{-1}(\frac{n_{1}}{n_{2}}\times sin\theta_{1})=sin^{-1}
(\frac{1.6}{1.4}\times sin55^{0})$
$\theta_{2}=sin^{-1}(1.14285\times 0.81915)=sin^{-1}(0.936165)$
$\theta_{2}=69.41^{0}$
refraction shown by the drawing is correct.
case (d) $n_{1}=1.6$ and $n_{2}=1.4$
angle of incidence $\theta_{1}=0^{0}$
$\theta_{2}=sin^{-1}(\frac{n_{1}}{n_{2}}\times sin\theta_{1})=sin^{-1}
(\frac{1.4}{1.6}\times sin0^{0})$
$\theta_{2}=sin^{-1}(0)$
$\theta_{2}=0^{0}$
refraction shown by the drawing is incorrect. because from calculations we are getting angle of refraction as $0^{0}$ whereas in drawing it is not zero.
