Question
What will be the path difference between two light waves $y_1=a_1 \sin \omega t$ and $y_2=a_2 \cos (\omega t+$ $\phi)$ ?
✓
Answer
Equation of first wave
$y_1=a_1 \sin \omega t$
Equation of second wave
$y_2=a_2 \cos (\omega t+\phi)$
or $y_2=\sin \left(\omega t+\phi+\frac{\pi}{2}\right)$
Phase difference between first and second wave
$\Delta \phi=\phi_2-\phi_1$
$=\left(\omega t+\phi+\frac{\pi}{2}\right)-\omega t$
or $\Delta \phi=\phi+\frac{\pi}{2}$
Phase difference due to path difference
$\Delta \phi=\frac{2 \pi}{\lambda} \times \Delta x$
Path difference
$\Delta x=\frac{\Delta \phi \times \lambda}{2 \lambda}$
$\therefore$ For given waves $I^{\text {st }}$ and $2^{\text {nd }}$, the path difference
$\Delta x=\frac{\lambda}{2 \pi}\left(\phi+\frac{\pi}{2}\right)$ $\qquad Ans. $
$y_1=a_1 \sin \omega t$
Equation of second wave
$y_2=a_2 \cos (\omega t+\phi)$
or $y_2=\sin \left(\omega t+\phi+\frac{\pi}{2}\right)$
Phase difference between first and second wave
$\Delta \phi=\phi_2-\phi_1$
$=\left(\omega t+\phi+\frac{\pi}{2}\right)-\omega t$
or $\Delta \phi=\phi+\frac{\pi}{2}$
Phase difference due to path difference
$\Delta \phi=\frac{2 \pi}{\lambda} \times \Delta x$
Path difference
$\Delta x=\frac{\Delta \phi \times \lambda}{2 \lambda}$
$\therefore$ For given waves $I^{\text {st }}$ and $2^{\text {nd }}$, the path difference
$\Delta x=\frac{\lambda}{2 \pi}\left(\phi+\frac{\pi}{2}\right)$ $\qquad Ans. $
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
An apple falls from a tree. An insect in the apple finds that the earth is falling towards it with an acceleration $g.$ Who exerts the force needed to accelerate the earth with this acceleration $g$ ?
→- Plot a graph showing variation of voltage vs the current drawn from the cell. How can one get information from this plot about the emf of the cell and its internal resistance?
- Two cells of emf’s $E_1$ and $E_2$ and internal resistance $r_1$ and $r_2$ are connected in parallel. Obtain the expression for the emf and internal resistance of a single equivalent cell that can replace this combination?
A load resistor of $2\text{k}\Omega$ is connected in the collector branch of an amplifier circuit using a transistor in common-emitter mode. The current gain $\beta=50.$ The input resistance of the transistor is $0.50\text{k}\Omega.$ If the input current is changed by $50\mu\text{A}.$
→- By what amount does the output voltage change?
- By what amount does the input voltage change?
- What is the power gain?
Consider a non-conducting plate of radius r and mass m that has a charge q distributed uniformly over it. The plate is rotated about its axis with an angular speed $\omega.$ Show that the magnetic moment $\mu$ and the angular momentum of the plate are related as $\mu=\frac{\text{q}}{2\text{m}}\text{l}.$
→A particle A having a charge of $2.0 \times 10^{-6}C$ is held fixed on a horizontal table. A second charged particle of mass $80g$ stays in equilibrium on the table at a distance of $10\ cm$ from the first charge. The coefficient of friction between the table and this second particle is $\mu=0.2.$ Find the range within which the charge of this second particle may lie.
→Consider the circuit arrangement shown in Fig $(a)$ for studying input and output characteristics of npn transistor in $CE$ configuration.
Select the values of $R_B$ and $R_C$ for a transistor whose $V_{BE} = 0.7V,$ so that the transistor is operating at point $Q$ as shown in the characteristics shown in Fig. $(b).$
Given that the input impedance of the transistor is very small and $V_{CC} = V_{BB} = 16V,$ also find the voltage gain and power gain of circuit making appropriate assumptions.
→Select the values of $R_B$ and $R_C$ for a transistor whose $V_{BE} = 0.7V,$ so that the transistor is operating at point $Q$ as shown in the characteristics shown in Fig. $(b).$
Given that the input impedance of the transistor is very small and $V_{CC} = V_{BB} = 16V,$ also find the voltage gain and power gain of circuit making appropriate assumptions.
White coherent light $(400\ nm - 700\ nm)$ is sent through the slits of a Young's double slit experiment $($figure.$)$ The separation between the slits is $0.5\ mm$ and the screen is $50\ cm$ away from the slits. There is a hole in the screen at a point $1.0\ mm$ away $($along the width of the fringes$)$ from the central line.
→- Which wavelength $(s)$ will be absent in the light coming from the hole?
- which wavelength $(s)$ will have a strong intensity?
The current−voltage characteristic of an ideal $p-n$ junction diode is given by $\text{i}=\text{i}_0\Big(\text{e}^{\frac{\text{eV}}{\text{kT}}}-1\Big)$ where, the drift current $i_0$ equals $10\mu\text{A}.$ Take the temperature $T$ to be $300K.$
→- Find the voltage $V_0$ for which $\text{e}^{\frac{\text{eV}}{\text{KT}}}=100.$ One can neglect the term $1$ for voltages greater than this value.
- Find an expression for the dynamic resistance of the diode as a function of $V$ for $V > V_0.$
- Find the voltage for which the dynamic resistance is $0.2\Omega.$
Explain electric charge. Verify by experiment that there is repulsion between like charges and attraction between unlike charges.
Radiation coming from transition $n = 2$ to $n = 1$ of hydrogen atoms falls on helium ions in $n = 1$ and $n = 2$ states. What are the possible transitions of helium ions as they absorbs energy from the radiation?
→