A system illustrated in the given figure 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 distance d, the radiation wavelength is equal to $\lambda$. Taking into account that the oscillations of source $2$ lag in phase behind the oscillations of source $1$ by $\phi$ ($\phi<\pi$), find the angles $\theta$ at which the radiation intensity is maximum:
This is how I solved:
From the figure given above the two waves interfering at P have covered different distances $S_1P=x$ and $S_2P=x+\Delta x$. The electric fields at P due to the two waves may be written as $$E_1=E_0\sin(kx-\omega t)$$ $$and\enspace \enspace ^*E_2=E_0\sin(k(x+\Delta x)-\omega t)$$ $^*$But we know that $E_2$ is initially behind $E_1$ by a phase $\phi$.So the actual equation of electric field at point P becomes: $$E_2=E_0\sin(k(x+\Delta x)-\omega t-\phi)$$ $$\implies E_2=E_0\sin(kx-\omega t+k\Delta x-\phi)$$ Hence the net phase difference at P will be $$\phi_{net}=(kx-\omega t+k\Delta x-\phi)-(kx-\omega t)$$ $\phi_{net}=k\Delta x -\phi$.But we the that $\Delta x=S_2A=d\sin(\frac{\pi}{2}-\theta)=d\cos(\theta)$ $$\phi_{net}=\left(\frac{2\pi }{\lambda}\right)d\cos(\theta)-\phi$$ For the intensity maxima $\phi_{net}=2n\pi$ $$\implies cos(\theta)=\frac{n+\frac{\phi}{2\pi}}{\frac{d}{\lambda}} $$ But my answer is slightly different from the answer provided in the text. I cannot figure out any flaw in my method. Following in the author's solution:
The phase difference produced due to the extra path is $\phi_1=\frac{2\pi}{\lambda}d\cos(\theta)$.
But the interfering waves has an additional phase difference given as $\phi_2=\phi$. Hence $\phi_{net}=\phi_1+\phi_2$ $$for\enspace maxima\enspace 2n\pi=\phi_1+\phi_2$$ $$\implies 2n\pi=\left(\frac{2\pi }{\lambda}\right)d\cos(\theta)+\phi $$ $$\implies cos(\theta)=\frac{n-\frac{\phi}{2\pi}}{\frac{d}{\lambda}} $$
Here the author's explanation does't seem logical to me as he has not considered the fact ray $S_2P$ is lagging behind the ray$S_1P$ and rather just added the two phase angles.
I would be glad to know if I am correct or wrong, with an explanation for it.