The refraction of light ray takes place from air to water, water to glass and again glass to air. The ray…
The refraction of light ray takes place from air to water, water to glass and again
glass to air. The ray emerges parallel to incident ray. The correct relation is
$\left[\mathrm{n}_{\mathrm{a}}, \mathrm{n}_{\mathrm{w}}, \mathrm{n}_{\mathrm{g}}\right.$ represent refractive indices of air, water and glass respectively.
1. Use Snell's Law:
At each interface, Snell's law applies:
$n_1 \sin i=n_2 \sin r$
where:
- $n_1$ is the refractive index of the medium where the ray originates,
- $n_2$ is the refractive index of the medium where the ray enters,
- $i$ and $r$ are the angles of incidence and refraction, respectively.
2. Parallel Emergent Ray: Since the emergent ray is parallel to the incident ray, the net deviation caused by the refractions at the interfaces is zero. Mathematically, this means:
$n_{a \rightarrow w} \cdot n_{w \rightarrow g} \cdot n_{g \rightarrow a}=1$
3. Substitute Refractive Indices:
The refractive indices at each interface are:
$n_{a \rightarrow w}=\frac{n_w}{n_a}, \quad n_{w \rightarrow g}=\frac{n_g}{n_w}, \quad n_{g \rightarrow a}=\frac{n_a}{n_g}$
Substituting these into the product:
$\frac{n_w}{n_a} \cdot \frac{n_g}{n_w} \cdot \frac{n_a}{n_g}=1$
4. Simplify:
The terms cancel out:
$\frac{n_w}{n_a} \cdot \frac{n_g}{n_w} \cdot \frac{n_a}{n_g}=1$
This confirms that the emergent ray is parallel to the incident ray.
Relationship Between Refractive Indices:
From the above analysis, the relationship becomes:
$w^a \cdot g^w=a^g$
where:
- $\boldsymbol{w}^a=\frac{n_w}{n_a}$,
- $\boldsymbol{g}^w=\frac{n_g}{n_w}$,
- $a^g=\frac{n_a}{n_g}$.
The correct option is:
$\text { 4. } w^a \cdot g^w=a^g$
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