Question

Light from two coherent sources of the same amplitude $A$and wavelength $\lambda $ illuminates the screen. The intensity of central maxima is ${I_0}$ . If the sources were incoherent, the intensity at the same point will be
A. $4{I_0}$
B. $10{I_0}$
C. ${I_0}$
D. $\dfrac{{{I_0}}}{2}$

Answer 1

Hint: Use the principle of superposition to find the intensity of the individual source in the first case. The resultant intensity of two in coherent sources can be calculated by directly adding the intensities of individual sources.

Complete step by step answer:
Coherent sources are those sources of light which emit waves that have the same frequency and zero or constant phase difference, while incoherent sources are those which emit waves that have random frequencies and phase differences.
In the first case, it is given that the sources are coherent which means that the phase angle,
$\phi = 0$
By the principle of superposition, we know that
${I_{res}} = {I_1} + {I_2} = 2\sqrt {{I_1}{I_2}} \cos \phi $
But,
It is given that the amplitude of both the sources are same which means that the intensity at is also same.
So, ${I_1} = {I_2}$
${I_0} = I + I + 2I\cos 0$
$
 {I_0} = 4I \\
 I = \dfrac{{{I_0}}}{4} \\
$
Now for the second case, it is given that the sources are now incoherent
So,
$
 {I_{res}} = {I_1} + {I_2} \\
 {I_{res}} = I + I \\
 {I_{res}} = 2I \\
$
Putting the value of $I$we get,
$
 {I_{res}} = 2\left( {\dfrac{{{I_0}}}{4}} \right) \\
 {I_{res}} = \dfrac{{{I_0}}}{2} \\
$

d) Is correct.

Additional information: The principle of superposition states that when two or more waves meet at a point, the resultant displacement at that point is equal to the sum of the displacements of the individual waves at that point
Whenever two waves having the same frequency travel with the same speed along the same direction in a specific medium, then they superpose and create an effect known as the interference of waves. In a situation where two waves having similar frequencies move with the same speed along opposite directions in a specific medium, then they superpose to produce stationary waves. Finally, when two waves having slightly varying frequencies travel with the same speed along the same direction in a specific medium, they superpose to produce beats.

Note: When the two individual waves are exactly in phase the result is large amplitude.
On the other hand, if the two individual waves are in opposite phases then the waves cancel each other.