Three waves of equal frequency having amplitudes `10mum`, `4mum`, `7mum` arrive at a given point with successive phase difference of `pi//2`, the amplitude of the resulting wave in `mum` is given by

Options

A. 4
B. 5
C. 6
D. 7

Answer

B

Explanation

The amplitude of the waves are `a_(1)=10mum,a_(2)=4mu m and a_(3)=7mu m` and phase difference between Ist and IInd wave is `(pi)/(2)` and that between Iind and IIIrd wave is `(pi)/(2)`.
Then, phase difference between Ist and IIIrd is `pi`.
Combining Ist with IIIrd, their resultant amplitude is given by
`A_(1)^(2)=a_(1)^(2)+a_(3)^(2)=2a_(1)a_(3) cos phi`
`or A_(1)=sqrt(10^(2)+7^(2)+2xx10xx7 cos pi)`
`=sqrt(100+49-140)=sqrt(9)=3 mu m`
Now, combining this with IInd wave, we have the resultant amplitude
`A^(2)=A_(1)^(2)+a_(2)^(2)+2A_(1)a_(2) "cos"(pi)/(2)`
`or A=sqrt(3^(2)+4^(2)+2xx3xx4xxcos90^(@))`
`=sqrt(9+16)=sqrt(25)=5mu m`