Question

(1) What are coherent sources? Why are they necessary for observing a sustained interference pattern?
(2) Lights from two independent sources are not coherent. Explain.

Hint:

Coherent means a constant phase difference. Independent sources have randomly changing phase, so no steady fringes form.

Answer 1

Step 1: Define coherent sources:
Two sources are coherent when they emit waves of the same frequency (wavelength) and the phase difference between the waves stays constant with time. They need not be in phase, only the difference must not change.

Step 2: Why coherence is needed for a sustained pattern:
At a point on the screen the two waves add with a phase difference \(\phi\). The resultant intensity is \(I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\phi\). If \(\phi\) at a point stays constant, that point keeps the same intensity, so fixed bright and dark fringes appear and last. If \(\phi\) changes at random, the cosine term averages to zero and the screen shows uniform intensity \(I_1 + I_2\). No fringes are seen.

Step 3: Why two independent sources are not coherent:
Light from an ordinary source comes from a huge number of atoms. Each atom emits a wave train for only about \(10^{-10}\) s, and the phase of a new train is random. In two independent sources, these phases change quickly and independently, so the phase difference between the two beams changes at random many times in a second. So the sources are not coherent and no interference pattern is seen.

Final Answer:
Coherent sources keep a constant phase difference, which gives steady fringes. Independent sources do not, so they cannot interfere. Fringe separation is 7.2 mm and the angular width is about \(0.34^{\circ}\). \[ \boxed{\beta = 7.2\ \text{mm},\quad \theta \approx 0.34^{\circ}} \]