Question

The ratio of the intensity at the centre of a bright fringe to the intensity at a point one-quarter of the distance between two fringes from the centre is :

• A. $2$
• B. $1/2$
• C. $4$
• D. $16$

Answer 1

The correct option is A $2$

Two waves of a single source having an amplitude $A$ interfere. The resulting amplitude
$A_r^2=A_1^2+A_2^2+2A_{1}A\cos \delta$
where $A_1=A_2=A$ and $\delta =$phase difference between the waves
$\Rightarrow l_r=l_1+l_2+2\sqrt{l_1{l}_2}\cos \delta$
When the maxima occurs at the center, $\delta =0$
$\Rightarrow l_{r_1}=41$ ...$\left(1\right)$
Since the phase difference between two successive fringes is $2\pi$, the phase difference between two points separated by a distance equal to one quarter of the distance between the two, successive fringes is equal to $\delta =\left(2\pi \right)\left(\frac{1}{4}\right)=\frac{\pi }{2}$ radian
$\Rightarrow {l^{\prime }}_{r_2}=41{\cos }^2\left(\frac{\pi /2}{2}\right)=21$ ...$\left(2\right)$
Using $\left(1\right)$ and $\left(2\right)$
$\frac{l_{r_1}}{l_{r_2}}=\frac{41}{21}=2$
Hence $\left(A\right)$ is correct.