A system illustrated in Fig. consists of two coherent point sources `1` and `2` located in a certain plane so that their dipole moments are oriented at right angles to that plane. The sources are separated by a disatnce `d`, the radiation wavelength is equal to `lambda`. Taking into account that the oscillations of source `I` by `varphi (varphi lt pi)`, find:
(a) the angles `theta` at which the radiation intensity is maximum,
(b) the conditions under which the radiation intensity in the direction `theta = pi` is maximum and in the opposite direction, minimum.

Explanation

With dipole moment `_|_` to plane there is no variation with `theta` of individual radiation amplitude. Then the intensity variation is due to interfernence only.
In the direction given by angle `theta `the phase difference is
`(2pi)/(lambda) (d cos theta) + varphi = 2k pi` fro maxima
Thus `d cos theta = (k -(varphi)/(2pi)) lambda`
`k = 0, +- 1, +- 2,......`
We have added `varphi` to `(2pi)/(lambda) d cos theta` because the extra path that the wave from `2` has to traved in going to `P` (as compared to `1`) makes it lag more than it already is (due to `varphi`).
(b) Maximum for `theta = pi` gives `-d = (k - (varphi)/(2pi)) lambda`
Minimum for `theta = 0` gives `d = (k' (varphi)/(2pi) + (1)/(2))lambda`
Adding we get `(k + k' -(varphi)/(pi) + (1)/(2)) lambda = 0`
This can be true only if `k' =- k, varphi = (pi)/(2)`
since `0 lt varphi lt pi`
Then `-d = (k - (1)/(4))lambda`
Here `k = 0., -1, -2 , -3,.....`
(Otherwise `R.H.S.` will become `+ve`).
Putting `k =- bar(k), bar(k) = 0, +1, +2, +3,.......`
`d = (bar(k) + (1)/(4)) lambda`.