Two sources of light, each giving an intensity \( I_0 \) on a screen, illuminate the same region of the screen. Consider the following statements:
(A) If the sources are coherent, the intensity at points where the phase difference is zero is \( 4I_0 \).
(B) If the sources are coherent, the intensity at points where the phase difference is \( \pi \) is zero.
(C) If the sources are coherent, the intensity at points where the phase difference is \( \pi/2 \) is \( I_0 \).
(D) If the sources are incoherent, the average intensity at all points is \( 2I_0 \).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
When two coherent sources, each of intensity \( I_0 \), meet at a point with a steady phase difference \( \phi \), the resultant intensity there is \( I=4I_0\cos^{2}(\phi/2) \). For incoherent sources the phase difference keeps changing rapidly, and only a time-averaged intensity is seen.
(A) If the sources are coherent, the intensity at points where the phase difference is zero is \( 4I_0 \). — True. At \( \phi=0 \) the factor \( \cos^{2}0 \) is 1, so \( I=4I_0 \). This is constructive interference, the brightest points of the pattern.
(B) If the sources are coherent, the intensity at points where the phase difference is \( \pi \) is zero. — True. At \( \phi=\pi \) the factor \( \cos(\pi/2) \) is zero, so the two waves arrive out of phase and the intensity is zero (destructive interference).
(C) If the sources are coherent, the intensity at points where the phase difference is \( \pi/2 \) is \( I_0 \). — False. Putting \( \phi=\pi/2 \) in the intensity relation gives:
\( \begin{aligned} I &= 4I_0\cos^{2}(\pi/4) \\ &= 4I_0\times\frac{1}{2} \\ &= 2I_0 \end{aligned} \)
(D) If the sources are incoherent, the average intensity at all points is \( 2I_0 \). — True. The phase difference changes so fast that \( \cos^{2}(\phi/2) \) averages to \( 1/2 \) at every point. The intensities of the two sources then add, giving a uniform \( 2I_0 \) with no fringes.
NCERT (2026-27):
Coherent and Incoherent Addition of Waves – “The amplitude of the resultant displacement is 2a cos (φ/2) and therefore the intensity at that point will be”
\( I=4I_0\cos^2(\phi/2) \qquad (10.11) \)
“If φ = 0, ± 2π, ± 4π,… […] we will have constructive interference leading to maximum intensity. On the other hand, if φ = ± π, ± 3π, ± 5π … […] we will have destructive interference leading to zero intensity.”
Coherent and Incoherent Addition of Waves – “… However, if the two needles do not maintain a constant phase difference, then the interference pattern will also change with time and, if the phase difference changes very rapidly with time, the positions of maxima and minima will also vary rapidly with time and we will see a ‘time-averaged’ intensity distribution. When this happens, we will observe an average intensity that will be given by”
\( I=2I_0 \qquad (10.12) \)
“at all points.”