Two harmonic waves of monochromatic light$\text{y}_{1} = \text{a} \cos\omega\text{t}\text{ and } \text{y}_{2} = \text{a} \cos(\omega\text{t} + \Phi)$
are superimposed on each other. Show that maximum intensity in interference pattern is four times the intensity due to each slit. Hence write the conditions for constructive and destructive interference in terms of the phase angle$\Phi$.
are superimposed on each other. Show that maximum intensity in interference pattern is four times the intensity due to each slit. Hence write the conditions for constructive and destructive interference in terms of the phase angle$\Phi$.
CBSE OUTSIDE DELHI - SET 2 SOUTH 2016
Download our app for free and get started
Resultant displacement $\text{y} = \text{y}_{1} + \text{y}_{2}$ $ = \text{a}[ \cos(\omega\text{t}) + \cos(\omega\text{t} + \phi)]$ $ = 2 \text{ a}\cos\big(\frac{\phi}{2}\big)\cos\big(\omega\text{t} +\frac{\phi}{2}\big)$ $\therefore$ amplitude of resultant wave $= 2 \text{ a}\cos(\frac{\phi}{2} )$$\therefore$ Intensity $ = 4 \text{I}_{o}\cos^{2}(\frac{\phi}{2}) , \text{where } \text{I}_{o} =\text{a}^{2}$ is the intensity of each harmonic wave
At the maxima, $\phi =\pm2\text{n}\pi\therefore\cos^{2}\frac{\phi}{2} = 1 $ At the maxima, $\text{I} = 4 \text{ I}_{o} = 4 \times\text{intensity due to one slit }$ $\text{I} = 4 \text{ I}_{o}\cos^{2}(\frac{\phi}{2})$ For constructive interference, I is maximum It is possible when $\cos^{2}(\frac{\phi}{2}) =1 ; \frac{\phi}{2} =\text{n}\pi; \phi = 2 \text{n}\pi$ For destructive interference, I is minimum, i.e, I=0 It is possible when $\cos^{2}(\frac{\phi}{2}) = 0 ; \frac{\phi}{2} = \frac{(2\text{n} - 1 )\pi}{2}; \phi = (2\text{n}\pm1 )\frac{\phi}{2}$.
At the maxima, $\phi =\pm2\text{n}\pi\therefore\cos^{2}\frac{\phi}{2} = 1 $ At the maxima, $\text{I} = 4 \text{ I}_{o} = 4 \times\text{intensity due to one slit }$ $\text{I} = 4 \text{ I}_{o}\cos^{2}(\frac{\phi}{2})$ For constructive interference, I is maximum It is possible when $\cos^{2}(\frac{\phi}{2}) =1 ; \frac{\phi}{2} =\text{n}\pi; \phi = 2 \text{n}\pi$ For destructive interference, I is minimum, i.e, I=0 It is possible when $\cos^{2}(\frac{\phi}{2}) = 0 ; \frac{\phi}{2} = \frac{(2\text{n} - 1 )\pi}{2}; \phi = (2\text{n}\pm1 )\frac{\phi}{2}$.
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1Prove that the power dissipated in an ideal resistor connected to an ac source is $\frac{\text{V}^2_{\text{eff}}}{\text{R}}.$View Solution
- 2View Solution
- Mention two properties of soft iron due to which it is preferred for making an electromagnet.
- State Gauss's law in magnetism. How is it different from Gauss's law in electrostatics and why?
- 3View SolutionAn inductor 200 mH, capacitor 500 µF, resistor 10Ω are connected in series with a100 V, variable frequency a.c. source. Calculate the
- Frequency at which the power factor of the circuit is unity.
- Current amplitude at this frequency.
- Q-factor.
- 4View Solution
- In Young's double slit experiment, two slits are 1 mm apart, and the screen is placed 1 m away from the slits. Calculate the fringe width when light of wavelength 500 nm is used.
- What should be the width of each slit in order to obtain 10 maxima of the double slits pattern within the central maximum of the single slit pattern?
- 5View SolutionState Biot – Savart law. Deduce the expression for the magnetic field at a point on the axis of a current carrying circular loop of radius ‘R’, distant ‘x’ from the centre. Hence write the magnetic field at the centre of a loop.
- 6Two capacitors of capacitance $10\mu\text{F}$ and $20 \mu\text{F}$ are connected in series with a 6 V battery. After the capacitors are fully charged, a slab of dielectric constant (K) is inserted between the plates of the two capacitors. How will the following be affected after the slab is introduced:View Solution
- The electric field energy stored in the capacitors.
- The charges on the two capacitors.
- The potential difference between the plates of the capacitors.
- 7An inductor-coil, a capacitor and an AC source of rms voltage 24V are connected in series. When the frequency of the source is varied, a maximum rms current of 6.0A is observed. If this inductor coil is connected to a battery of emf 12V and internal resistance $4.0\Omega,$ what will be the current?View Solution
- 8A resistor of 100 $\Omega$ and a capacitor of $100 / \pi\mu\text{F}$ are connected in series to a 220 V, 50 Hz a.c. supply.View Solution
- Calculate the current in the circuit.
- Calculate the (rms) voltage across the resistor and the capacitor. Do you find the algebraic sum of these voltages more than the source voltage? If yes, how do you resolve the paradox?
- 9View Solution
- Find the value of the phase difference between the current and the voltage in the series LCR circuit shown below. Which one leads in phase: current or voltage?
- Without making any other change, find the value of the additional capacitor $C_1,$ to be connected in parallel with the capacitor C, in order to make the power factor of the circuit unity.
- 10View Solution
- When an AC source is connected to an ideal inductor show that the average power supplied by the source over a complete cycle is zero.
- A lamp is connected in series with an inductor and an AC source. What happens to the brightness if the lamp when the key is plugged in and an iron rod is inserted inside the inductor? Explain.
