Which of the following conditions is correct for the constructive interference?
(ΔX represents path difference, Δϕ represents phase difference, n is an integer)
Understanding Constructive Interference Conditions
Constructive interference occurs when two or more waves overlap in such a way that their resultant amplitude is greater than the amplitude of the individual waves. This happens when the waves are in phase.
Path Difference for Constructive Interference
For constructive interference, the path difference ($\Delta X$) between two waves must be an integer multiple of the wavelength ($\lambda$). This means that the waves travel distances that differ by whole numbers of wavelengths. The condition is represented as:
$$ \Delta X = n\lambda $$
Where '$n$' is an integer ($n = 0, 1, 2, 3, ...$).
- If $n=0$, the path difference is zero (waves are in phase).
- If $n=1$, the path difference is one wavelength.
- If $n=2$, the path difference is two wavelengths, and so on.
Phase Difference for Constructive Interference
The phase difference ($\Delta \phi$) is directly related to the path difference. When the path difference is an integer multiple of the wavelength ($n\lambda$), the phase difference is an even integer multiple of $\pi$ radians. The condition is:
$$ \Delta \phi = 2n\pi $$
Where '$n$' is an integer ($n = 0, 1, 2, 3, ...$).
- If $n=0$, the phase difference is $0$ radians (waves are perfectly in phase).
- If $n=1$, the phase difference is $2\pi$ radians (equivalent to being in phase).
- If $n=2$, the phase difference is $4\pi$ radians (also equivalent to being in phase).
Analyzing the Options
Let's examine each option based on the conditions for constructive interference:
- Option 1: $\Delta X = (2n+1)\lambda/2$ & $\Delta \phi = (2n+1)\pi$. This represents an odd path difference (half wavelengths) and an odd phase difference, which corresponds to destructive interference.
- Option 2: $\Delta X = n\lambda$ & $\Delta \phi = (2n + 1) \pi$. The path difference condition is correct, but the phase difference condition is incorrect for constructive interference.
- Option 3: $\Delta X = n\lambda$ & $\Delta \phi = 2n\pi$. Both the path difference ($n\lambda$) and the phase difference ($2n\pi$) conditions are correct for constructive interference.
- Option 4: $\Delta X = (2n+1)\lambda/2$ & $\Delta \phi = 2n\pi$. The phase difference condition is correct, but the path difference condition is incorrect for constructive interference.
- Option 5: This option is empty.
Conclusion
Based on the analysis, the correct conditions for constructive interference are a path difference of $n\lambda$ and a phase difference of $2n\pi$. Therefore, option 3 correctly states these conditions.
Important Questions from Interference
- A single slit of width $a$ is illuminated by a monochromatic light of wavelength $\lambda_1 = 6000 \text{ Å}$. The angular width of the central maximum observed in the Fraunhofer diffraction pattern is $\theta_1$. When the slit width is increased by $20\%$ and the light source is replaced with another monochromatic light of wavelength $\lambda_2$, the angular width of the central maximum becomes $\frac{3}{5}$ of its initial value, $\theta_1$. Determine the wavelength $\lambda_2$.
A system of three polarizers $P_1$, $P_2$, $P_3$ is set up such that the pass axis of $P_3$ is crossed with respect to that of $P_1$.
The pass axis of $P_2$ is inclined at $15^\circ$ to the pass axis of $P_1$.
When a beam of unpolarized light of intensity $I_0$ is incident on $P_1$, the intensity of light transmitted by the three polarizers is $I$. The ratio $(I_0/I)$ equals (nearly):The interference pattern is obtained with two coherent light sources. If the ratio of their amplitudes is $n$, then in the interference pattern, the ratio $\frac{{{I_{max}} - {I_{min}}}}{{{I_{max}} + {I_{min}}}}$ will be
- Two identical coherent waves are superimposed at a point. If the maximum possible resultant intensity from their interference is $I_{max}$, and the resultant intensity at this point is $I_{max}/4$, then find the phase difference between the two waves at this point.
Which of the following sources gives best monochromatic light?
