Question

In Fig. 35-37, two radio frequency point sources ${\mathit{S}}_{\mathbf{1}}$and ${\mathit{S}}_{\mathbf{2}}$, separated by distance $\mathit{d}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{}\mathit{m}$, are radiating in phase with $\mathit{\lambda }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{50}\mathbf{}\mathit{m}$. A detector moves in a large circular path around the two sources in a plane containing them. How many maxima does it detect?

Short answer

The total number of the maxima is 16.

Step-by-step solution

Write the given data from the question

The distance between the slits, $d=2\text{m}$.

The wavelength, $\lambda =0.50\text{m}$.

Determine the formulas to calculate the number of the maxima detect by the detector

The condition for the maxima in Young’s experiment is given as follows.

$\mathit{d}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathbf{=}\mathit{m}\mathit{\lambda }$ …… (1)

Here,d is the distance between the slits, $\mathit{\lambda }$is the wavelength, mis the order, and$\mathit{\theta }$is the angular separation.

Calculate the number of the maxima detect by the detector

Calculate the number of the maxima.

Substitute $0.50\text{m}$for $\lambda$and $2\text{m}$ for d into equation (1).

$$\begin{gathered} 2\sin \theta =m\times 0.50 \\ \sin \theta =\frac{1}{2}\times m\times 0.50 \\ \sin \theta =\frac{1}{4}\times m \end{gathered}$$

Now,

$$\begin{gathered} \left|\sin \theta \right|⩽1 \\ \left|\frac{m}{4}\right|⩽1 \end{gathered}$$.

The value of m can be negative as well as positive. Therefore, m will have $-4,-3,-2,-1,0,+1,+2,+3,+4$.

For each of the value there is two values of the $\theta$.

Therefore, one value for $-{90}^{0}C$and other for $+{90}^{0}C$.

Hence, the total number of the maxima is 16.