i)   Discuss Doppler’s effect in electromagnetic waves. 

ii)  Consider two coherent sources ${\mathrm{S}}_1 \mathrm{and} {\mathrm{S}}_2$producing monochromatic waves to produce an interference pattern. Let the displacement of the wave produced by ${\mathrm{S}}_1$be given by ${\mathrm{y}}_1=\mathrm{acoswt}$ and the displacement by${\mathrm{S}}_2$ be ${\mathrm{y}}_2=\mathrm{acos}(\mathrm{wt}+\mathrm{\varphi })$.Find out the expression for the amplitude of the resultant displacement at a point and show that the intensity at that point will be Hence, establish the conditions for constructive and destructive interference.

                                                          OR

i) A ray PQ of light is incident on the face AB of a glass prism ABC (as shown in the figure) and emerges out of the face AC. Trace the path of therapy. Show that,
$\mathrm{i}+\mathrm{e}=\mathrm{A}+\mathrm{\delta }$

 

 

 

 

 

where, d and e denote the angle of deviation and angle of emergence, respectively. Plot a graph showing the variation of the angle of deviation as a function of angle of incidence. State the condition under which $\angle \mathrm{\delta }$is minimum. 

(ii) Find out the relation between the refractive index $\mathrm{\mu }$ of the glass prism and $\angle \mathrm{A}$ for the case, when the angle of prism A is equal to the angle of minimum deviation ${\mathrm{\delta }}_{\mathrm{m}}$ . Hence, obtain prism A of the refractive index for angle of prism A = 60°

i)According to this effect, whenever there is a relative motion between a source of light and observer, the apparent frequency of light received by observer is different from the true frequency of light emitted from the source of light. Astronomers call the increase in wavelength due to Doppler effect as red shift, since a wavelength in the middle of the visible region of spectrum moves towards the red end of the spectrum. When waves are received from a source moving towards the observer, there is an apparent decrease in wavelength, this is referred to as blue shift. The fractional change in frequency is given by

$\frac{∆\mathrm{v}}{\mathrm{v}}=\frac{{\mathrm{\nu }}_{\mathrm{radial}}}{\mathrm{c}}$

ii) $$\begin{gathered} \mathrm{Given}, \mathrm{the} \mathrm{displacements} \mathrm{of} \mathrm{two} \mathrm{coherent} \mathrm{sources} {\mathrm{y}}_1=\mathrm{a} \cos \\ \mathrm{\omega t} \mathrm{and} {\mathrm{y}}_2=\mathrm{a} \cos (\mathrm{\omega t}+\mathrm{\varphi }) \\ \mathrm{By} \mathrm{principle} \mathrm{of} \mathrm{superposition}, \\ \mathrm{y}={\mathrm{y}}_1+{\mathrm{y}}_2=\mathrm{a} \cos \mathrm{\omega t}+\mathrm{a} \cos (\mathrm{\omega t}+\mathrm{\varphi }) \\ \mathrm{y}=\mathrm{a} \cos \mathrm{\omega t}+\mathrm{a} \cos \mathrm{\omega t} \mathrm{cos\varphi }-\mathrm{asin} \mathrm{\omega t} \sin \mathrm{\varphi } \\ \mathrm{y}=\mathrm{a}(1+\cos \mathrm{\varphi })\cos \mathrm{\omega t}+(-\mathrm{asin} \mathrm{\varphi })\sin \mathrm{\omega t} \\ \mathrm{Let} \mathrm{a}(1+\mathrm{cos\varphi })=\mathrm{Acos} \mathrm{\theta } \\ \mathrm{and} \mathrm{asin} \mathrm{\varphi } =\mathrm{A} \sin \mathrm{\theta } \\ \therefore \mathrm{y} =\mathrm{A} \mathrm{cos\theta } \mathrm{cos\omega t}-\mathrm{A} \sin \mathrm{\theta sin} \mathrm{\omega t} \\ \Rightarrow \mathrm{y} =\mathrm{A} \cos (\mathrm{\omega t}+\mathrm{\theta }) \\ \mathrm{Squaring} \mathrm{and} \mathrm{adding} \mathrm{Eqs}. \left(\mathrm{i}\right) \mathrm{and} \left(\mathrm{ii}\right), \mathrm{we} \mathrm{get} \\ {(\mathrm{A} \cos \mathrm{\theta })}^2+{(\mathrm{A} \sin \mathrm{\theta })}^2={\mathrm{a}}^2{(1+\cos \mathrm{\varphi })}^2+{(\mathrm{a} \sin \mathrm{\varphi })}^2 {\mathrm{A}}^2({\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }) \\ ={\mathrm{a}}^2(1+{\cos }^2\mathrm{\varphi }+2\mathrm{cos\varphi })+{\mathrm{a}}^2 {\sin }^2\mathrm{\varphi } \\ \Rightarrow {\mathrm{A}}^2\times 1={\mathrm{a}}^2+{\mathrm{a}}^2+2{\mathrm{a}}^2 \mathrm{cos\varphi }=2{\mathrm{a}}^2(1+\cos \mathrm{\varphi }) \\ \Rightarrow {\mathrm{A}}^2=2{\mathrm{a}}^2\left(2 {\cos }^2\frac{\mathrm{\varphi }}{2}\right)=4{\mathrm{a}}^2 {\cos }^2\left(\frac{\mathrm{\varphi }}{2}\right) \\ \mathrm{If} \mathrm{I} \mathrm{is} \mathrm{the} \mathrm{resultant} \mathrm{intensity}, \mathrm{then} \mathrm{I}=4{\mathrm{a}}^2 {\cos }^2\frac{\mathrm{\varphi }}{2}. \end{gathered}$$

$$\begin{gathered} \mathrm{From} \mathrm{constructive} \mathrm{interference}, \cos \frac{\mathrm{\varphi }}{2}=\pm 1 \\ \Rightarrow \frac{\mathrm{\varphi }}{2}=2\mathrm{\pi }\Rightarrow \mathrm{\varphi }=2\mathrm{n\pi } \\ \mathrm{For} \mathrm{destructive} \mathrm{interference}, \\ \cos \frac{\mathrm{\varphi }}{2}=0 \\ \Rightarrow \frac{\mathrm{\varphi }}{2}=(2\mathrm{n}+1)\frac{\mathrm{\pi }}{2} \\ \mathrm{\varphi }=(2\mathrm{n}+1)\mathrm{\pi } \end{gathered}$$

OR

$$\begin{gathered} \mathrm{Let} \mathrm{PQ} \mathrm{and} \mathrm{RS} \mathrm{are} \mathrm{incident} \mathrm{and} \mathrm{emergent} \mathrm{rays} \mathrm{and} \\ \mathrm{incident} \mathrm{ray} \mathrm{get} \mathrm{deviated} \mathrm{by} \mathrm{\delta } \mathrm{by} \mathrm{the} \mathrm{prism}. \\ \mathrm{i}.\mathrm{e}., \angle \mathrm{TMS}=\mathrm{\delta } \\ \mathrm{Let} {\mathrm{\delta }}_1 \mathrm{and} {\mathrm{\delta }}_2 \mathrm{are} \mathrm{deviation} \mathrm{produced} \mathrm{at} \mathrm{refractions} \mathrm{taking} \\ \mathrm{place} \mathrm{at} \mathrm{AB} \mathrm{and} \mathrm{AC}, \mathrm{respectively} \end{gathered}$$

$$\begin{gathered} \therefore \mathrm{\delta }={\mathrm{\delta }}_1+{\mathrm{\delta }}_2=(\mathrm{i}-{\mathrm{r}}_1)+(\mathrm{e}-{\mathrm{r}}_2) \\ =(\mathrm{i}+\mathrm{e})-({\mathrm{r}}_1+{\mathrm{r}}_2) \dots \left(\mathrm{i}\right) \\ \mathrm{But} \mathrm{in} ∆\mathrm{FNR}, \\ \angle \mathrm{QNR}+\angle \mathrm{RQN}+\angle \mathrm{QRN}=180° \\ \mathrm{or} \angle \mathrm{QNR}=180°-({\mathrm{r}}_1+{\mathrm{r}}_2) \dots \left(\mathrm{ii}\right) \\ \mathrm{In} \square \mathrm{QARN}, \angle \mathrm{AQN} \mathrm{and} \angle \mathrm{ARN} \mathrm{are} \mathrm{right} \mathrm{angles}. \\ \mathrm{So}, \angle \mathrm{QNR}=180°-\mathrm{A} \dots \left(\mathrm{iii}\right) \\ \mathrm{where}, \mathrm{A} \mathrm{is} \mathrm{angle} \mathrm{of} \mathrm{prism}. \\ \mathrm{From} \mathrm{Eqs}. \left(\mathrm{ii}\right) \mathrm{and} \left(\mathrm{iii}\right), \mathrm{we} \mathrm{have} \\ \mathrm{A}={\mathrm{r}}_1+{\mathrm{r}}_2 \dots \left(\mathrm{iv}\right) \\ \mathrm{From} \mathrm{Eqs}. \left(\mathrm{i}\right) \mathrm{and} \left(\mathrm{iv}\right), \mathrm{we} \mathrm{have} \\ \mathrm{\delta }=(\mathrm{i}+\mathrm{e})-\mathrm{A} \dots \left(\mathrm{v}\right) \\ \mathrm{or} \mathrm{i}+\mathrm{e}=\mathrm{A}+\mathrm{\delta } \\ \left(\mathrm{ii}\right) \mathrm{i}-\mathrm{\delta } \mathrm{graph} \mathrm{is} \mathrm{shown} \mathrm{in} \mathrm{the} \mathrm{figure} \end{gathered}$$

$$\begin{gathered} \mathrm{The} \mathrm{conditions} \mathrm{for} \mathrm{the} \mathrm{angle} \mathrm{of} \mathrm{minimum} \mathrm{deviation} \mathrm{are} \mathrm{given} \mathrm{as} \mathrm{below}. \\ \left(\mathrm{a}\right) \mathrm{Angle} \mathrm{of} \mathrm{incidence} \mathrm{i} \mathrm{and} \mathrm{angle} \mathrm{of} \mathrm{emergence} \mathrm{e} \mathrm{are} \mathrm{equal}. \\ \mathrm{i}.\mathrm{e}., \angle \mathrm{r}=\angle \mathrm{e} \\ \left(\mathrm{b}\right) \mathrm{In} \mathrm{equilateral} \mathrm{prism}, \mathrm{the} \mathrm{refracted} \mathrm{ray} \mathrm{is} \mathrm{parallel} \mathrm{to} \mathrm{base} \mathrm{of} \mathrm{prism}. \\ \left(\mathrm{c}\right) \mathrm{The} \mathrm{incident} \mathrm{and} \mathrm{emergent} \mathrm{rays} \mathrm{are} \mathrm{bent} \mathrm{on} \mathrm{same} \mathrm{angle} \mathrm{from} \mathrm{refracting} \\ \mathrm{surfaces} \mathrm{of} \mathrm{the} \mathrm{prism}. \\ \mathrm{i}.\mathrm{e}. \angle {\mathrm{r}}_1=\angle {\mathrm{r}}_2 \\ \mathrm{For} \mathrm{minimum} \mathrm{deviation} \mathrm{position}, \\ \mathrm{Putting} \mathrm{r}={\mathrm{r}}_1={\mathrm{r}}_2 \mathrm{and} \mathrm{i}=\mathrm{e} \mathrm{in} \mathrm{Eq}. \left(\mathrm{iv}\right) \\ 2\mathrm{r}=\mathrm{A}\Rightarrow \mathrm{r}=\frac{\mathrm{A}}{2} \dots \left(\mathrm{vi}\right) \\ \mathrm{From} \mathrm{Eq}. \left(\mathrm{i}\right), {\mathrm{\delta }}_{\mathrm{m}}=2\mathrm{i}-\mathrm{A} \\ \mathrm{i}=\frac{\mathrm{A}+{\mathrm{\delta }}_{\mathrm{m}}}{2} \dots \left(\mathrm{vii}\right) \\ \therefore \mathrm{Refractive} \mathrm{index} \mathrm{of} \mathrm{material} \mathrm{of} \mathrm{prism}, \\ \mathrm{\mu }\frac{\sin \mathrm{i}}{\sin \mathrm{r}} \\ \mathrm{From} \mathrm{Eqs}. \left(\mathrm{vi}\right) \mathrm{and} \left(\mathrm{vii}\right), \mathrm{we} \mathrm{get} \\ \Rightarrow \mathrm{\mu }=\frac{\sin \left({\displaystyle \frac{\mathrm{A}+{\mathrm{\delta }}_{\mathrm{m}}}{2}}\right)}{\sin (\mathrm{A}/2)} \\ \mathrm{i}) \mathrm{As} \mathrm{per} \mathrm{the} \mathrm{question}, \\ \mathrm{Angle} \mathrm{of} \mathrm{prism}, \mathrm{A}=\mathrm{Angle} \mathrm{of} \mathrm{minimum} \mathrm{deviation}, {\mathrm{\delta }}_{\mathrm{m}} \\ \mathrm{i}.\mathrm{e}. \angle \mathrm{A}=\angle {\mathrm{\delta }}_{\mathrm{m} \dots \left(\mathrm{viii}\right)} \\ \mathrm{Substituting} \mathrm{the} \mathrm{value} \mathrm{of} \angle {\mathrm{\delta }}_{\mathrm{m}} \mathrm{from} \mathrm{Eq}. (\mathrm{viii}) \mathrm{we} \mathrm{get} \\ \Rightarrow \mathrm{\mu }=\frac{\sin \left({\displaystyle \frac{\mathrm{A}+\mathrm{A}}{2}}\right)}{\sin (\mathrm{A}/2)} \\ \Rightarrow \mathrm{\mu }=\frac{\sin \mathrm{A}}{\sin (\mathrm{A}/2)} \\ \Rightarrow \mathrm{\mu }=\frac{2\sin (\mathrm{A}/2).\cos (\mathrm{A}/2)}{\sin (\mathrm{A}/2)} \\ \Rightarrow \mathrm{\mu }=2\cos (\mathrm{A}/2) \\ \mathrm{This} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{relation} \mathrm{between} \mathrm{refractive} \mathrm{index} \mathrm{of} \mathrm{the} \mathrm{glass} \\ \mathrm{prism} \mathrm{and} \mathrm{angle} \mathrm{of} \mathrm{prism}. \\ \mathrm{Since}, \angle \mathrm{A}=60° (\mathrm{given}) \\ \Rightarrow \mathrm{\mu }=2 \cos \left(\frac{60°}{2}\right) \\ \Rightarrow \mathrm{\mu }=2 \cos 30° \\ \Rightarrow \mathrm{\mu }=2\times \frac{\sqrt{3}}{2} \\ \Rightarrow \mathrm{\mu }=\sqrt{3} \\ \Rightarrow \mathrm{\mu }=1.732 \end{gathered}$$