Question

For a prism of refracting angle A and refractive index 2. Assume rays are incident at all angles of incidence $0^o\le i\le {90}^o$. Ignore partial reflection

Answer 1

Using $\sin {\theta }_c=\frac{1}{n_2}$

$⟹{\theta }_c={30}^o$ (As $n_2=2$)

Here ${\theta }_c=r_2$

This suggests that $-$

1) if $r_2={30}^o,$ then the ray will graze on the face LN of the prism.

2) if $r_2<{30}^o$, then the ray will always be refracted from the face LN.
3) if $r_2>{30}^o$, then always total internal reflection will occur.

Using snell' law for face LM, $1\times sini=2sin{r}_1$

Thus for $0^o\le i\le {90}^o,⟹0^o\le r_1\le {30}^o$

As $A=r_1+r_2$

Now for Case A; $A={15}^o$

$r_1+r_2={15}^o⟹0^o\le r_2\le {15}^o$

Hence all the rays will refract into the air.

For Case B; $A={45}^o$

$r_1+r_2={45}^o⟹{15}^o\le r_2<{45}^o$

Hence some rays will refract and some will reflect.

For Case C; $A={70}^o$

$r_1+r_2={70}^o⟹{40}^o\le r_2<{70}^o$

Hence all the rays will be reflected back from face LN.

For Case D; $A={50}^o$

$r_1+r_2={50}^o⟹{20}^o\le r_2<{50}^o$

Hence some rays will refract and some will reflect.