Question

Algebraically sum the two electric field waves

E1=E01cos([omega]t + [alpha]1) and E2=E02([omega]t +[alpha]2)

under the condtions E02=E01 and aplha1=0. Your result should be a single harmonic (sine or cosine) function with appropriate amplitude and phase. Express the amplitude and phase in terms of parameters of the two combined waves.

Answer 1

Electric fields,

$E1=E01\cos(\omega t+\alpha _{1})$

And,

$E2=E02\cos(\omega t+\alpha _{2})$

The angular frequency of the two electric field are same.

The amplitude of the two electric fields are,

$E_{sum}=\sqrt{(E01)^{2}+(E02)^{2}}$

It is given that,

$E02=E01$

So, the sum of two electric fields amplitude are,

$E_{sum}=\sqrt{(E01)^{2}+(E01)^{2}}$

$E_{sum}=\sqrt{2E01^{2}}$

${\color{Blue} E_{sum}=\sqrt{2}E01}$

The value of phase is,

$\alpha=\alpha _{1}+\alpha _{2}$

it is given that,

$\alpha _{1}=0$

therefore, the value of phase angle is,

$\alpha =0+\alpha _{2}$

${\color{Blue} \alpha =\alpha _{2}}$

Express the amplitude and phase in terms of parmeters of the two combined waves is,

${\color{Blue} E_{sum}=\sqrt{2}E01(\cos(\omega t+\alpha 2))}$