Question

Find the radius of the nth bright ring.

• A. $D\sqrt{1(1-\frac{n\lambda }{d})}$
• B. $D\sqrt{2(1-(m-n\left)\frac{\lambda }{d}\right)}$
• C. $2D\sqrt{2(1-\frac{n\lambda }{d})}$
• D. $D\sqrt{2(1-\frac{n\lambda }{2d})}$

Answer 1

Answer: (B)

$D\sqrt{2(1-(m-n\left)\frac{\lambda }{d}\right)}$

For center point O to be bright spot, then $d=m\lambda$

Let the radius of $n^{th}$ bright ring to be obtained at a point P be $r$.

Path difference between the light reaching point P, $\Delta x=dcos\theta =(m-n)\lambda$

As $\theta$ is very small, so we can aprproximate $cos\theta \approx 1-\frac{{\theta }^2}{2}$. Also similarly $\theta \approx sin\theta \approx \frac{r}{D}$ for $D>>d$

Putting these values in the above equation, $(1-\frac{r^2}{2D^2})=(m-n)\frac{\lambda }{d}$

Thus radius, $r=D\sqrt{2(1-(m-n\left)\frac{\lambda }{d}\right)}$