Q.(a) State the conditions for total internal reflection to take place.

Concept: Refraction at a plane surface — apparent depth depends on the refractive index of the medium.

(a) Conditions for total internal reflection:

1. Light must travel from a denser to a rarer medium.
2. The angle of incidence in the denser medium must exceed the critical angle for the pair of media.

(b) (i) Graph of apparent depth vs real depth:

For a liquid of refractive index $n$, apparent depth $d_a = \frac{d_r}{n}$, where $d_r$ is the real depth measured from the surface. This is a straight line through the origin with slope $1/n$.

(a) Conditions for total internal reflection

Total internal reflection (TIR) is a phenomenon where light is completely reflected back into the original medium, with no transmission into the second medium. This happens only when three conditions are simultaneously satisfied:

1. Light must travel from a denser to a rarer medium — that is, from a medium of higher refractive index ($n_1$) to one of lower refractive index ($n_2$). For example, from glass to air, or from water to air.

2. The angle of incidence must exceed the critical angle for the pair of media. The critical angle $i_c$ is defined by:

where $n_1 > n_2$. When $i > i_c$, Snell’s law ($n_1 \sin i = n_2 \sin r$) would require $\sin r > 1$, which is impossible — so no refraction occurs, and the ray is totally reflected.

3. The incident ray must be in the denser medium — TIR cannot happen when light goes from a rarer to a denser medium, because the refracted ray always bends toward the normal, never away.

(b) Apparent depth in a liquid

When you look straight down into a liquid, objects appear shallower than they really are. This is because light from the object bends away from the normal as it leaves the liquid (denser) into air (rarer). For near-normal viewing (small angles), the relation is simple:

Here $n$ is the refractive index of the liquid relative to air (taking air as $n_{\text{air}} \approx 1$). This formula assumes the observer is directly above the object, so the line of sight is nearly vertical.

(i) Plotting the graph

Let’s denote:

• Real depth of the coin = $d$ (measured from the liquid surface down to the coin)
• Apparent depth = $d_a$

From the formula: $d_a = \frac{1}{n} \cdot d$

This is a linear equation of the form $y = mx$, where:

• $y = d_a$
• $x = d$
• Slope $m = 1/n$

So the graph is a straight line passing through the origin. As the coin is lowered from the surface ($d=0$) to the bottom ($d=H$), the apparent depth increases proportionally.

The graph is a straight line from $(0,0)$ to $(H, H/n)$. …