Define wavefront of a travelling wave. Using Huygens principle, obtain the law of refraction at a plane interface when light passes from a denser to rarer medium.
CBSE 55-1-3 PAPER SET 2020
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A plane wavefront $A B$ is incident in rarer medium at instant $t=0$ on interface $X Y$ separating it from a denser medium. When wavelet $A$ is on interface, $B$ is at a distance $B B$, from it. It takes $t$ time to cover the distance $B B_1=v_1 t$ $=$ to reach on interface $X Y$. Mean while, the wavelet from $A$ reaches to point $A_1$ covering a distance $A A_1=v_2 t$ in denser medium.
To locate $A_1$, draw a secondary wavelet with radius $A_1=v_2 t \&$ centre $A$. Draw tangent from $B$, into this sec. wavelet intersecting at $\mathrm{A}_1$.
$A_1 B_1$ is refracted wavefront at instant $t$.
$\mathrm{i}=$ angle of incidence,
$r=$ angle of refraction.
$\therefore\triangle\text{ABB}_1$
$\Rightarrow\sin\text{i}=\frac{\text{BB}_1}{\text{AB}_1}$
$\triangle\text{AA}_1\text{B}_1$
$\Rightarrow\sin\text{r}=\frac{\text{AA}_1}{\text{AB}_1}$
$\therefore\frac{\sin\text{i}}{\sin\text{r}}=\frac{\text{BB}_1}{\text{AA}_1}=\frac{\text{v}_1\text{t}}{\text{v}_2\text{t}}=\frac{\text{v}_1}{\text{v}_2}$
$\therefore\frac{\sin\text{i}}{\sin\text{r}}=\frac{\text{v}_1}{\text{v}_2}=\text{constant}$
Also, $\text{n}_1=\frac{\text{c}}{\text{v}_1}\text{n}_2=\frac{\text{c}}{\text{v}_2}$
$\frac{\text{n}_1}{\text{n}_2}=\frac{\text{v}_2}{\text{v}_1}$
$\Rightarrow\frac{\sin\text{i}}{\frac{\sin\text{r}}{}}=\frac{\text{n}_2}{\text{n}_1}=\text{constant}$
Which is Snell's law.
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