Question

When a sigle slit is replaced by the double slit arrangement, the intensity of central maximum is increased by?

Answer 1

Since,

$$I_{double \:slit}=I_o\cos^2(\frac{\Delta \phi}{2})$$

Where,

$$\Delta \phi=\frac{2\pi}{\lambda}d\sin(\theta)$$

Now, Central maximum occurs when $d\sin(\theta)=\lambda$ ,thus from above formula we get,

$$I_{double \:slit}=I_o\cos^2(\frac{2\pi}{2})=I_o\hspace{1cm}(1)$$

And

$$I_{single \: slit}=I_o\bigg(\frac{\sin(\beta)}{\beta}\bigg)^2$$

where,

$$\beta=\frac{\pi}{\lambda}a\sin(\theta)$$

Thus, we have central maximum when,$\theta\rightarrow 0\implies \beta\rightarrow0$ ,thus$\frac{\sin(\beta)}{\beta}\rightarrow 1$ ,

Hence,

$$I_{single \: slit}=I_o\hspace{1cm}(2)$$

Therefore, from equations (1) and (2) we get,

Intensity for both the slit at central maximum is equal.