Three light waves combine at a point find resultant amplitude and phase angle

In summary, the problem involves three light waves with different electric field components that combine at a point. The resultant amplitude and phase angle of the electric field at that point need to be determined. The solution involves using double angle formulas to add all three waves at once and then simplifying to get the final result of ER = Eo sin(\omegat).
  • #1
CianM
3
0

Homework Statement


Three light waves combine at a point where their electric field components are

E1 = Eo[tex] sin \omega[/tex]t

E2 = Eo[tex] sin (\omega[/tex]t - 2[tex]\pi[/tex]/3)

E3 = Eo[tex] sin (\omega[/tex]t + [tex]\pi[/tex]/3)

Find the resultant amplitude of the electric field ER at that point and it's phase angle[tex]\beta[/tex]
Write the resultant wav int the form E = ER[tex] sin(\omega[/tex]t + [tex]\beta[/tex])

Homework Equations





The Attempt at a Solution



Am I right in assuming that first you add E1+E2 then add E12 + E3 using double angle formulas? Or am I going about this completely the wrong way?
 
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  • #2
What about adding all three at once, using dbl angle formula to break out the phase shifts and collecting like terms. The results should be the same.
 
  • #3
Well I already did the way I suggested and the answer I got was :
ER = 2Eo sin ([tex]\omega[/tex]t)cos([tex]\pi[/tex]/3)
taking cos([tex]\pi[/tex]/3) = 1/2
then equals ER = Eosin( [tex]\omega [/tex]t) .
Is this right?
 
  • #4
yep, some careful examination of the problem shows that the -phase shift term is equal and opposite to the positive shift term, cancelling out, leaving your result. (in other words the two phase angles sum to pi)
 
Last edited:
  • #5
Ok. Thanks for your help!
 

1. How do you calculate the resultant amplitude of three light waves at a point?

The resultant amplitude of three light waves at a point can be calculated using the formula: A = √(A1² + A2² + A3² + 2A1A2cos(ϕ1 - ϕ2) + 2A1A3cos(ϕ1 - ϕ3) + 2A2A3cos(ϕ2 - ϕ3)), where A1, A2, and A3 are the amplitudes of the three individual waves and ϕ1, ϕ2, and ϕ3 are their respective phase angles.

2. What is the phase angle of the resultant light wave?

The phase angle of the resultant light wave is calculated using the formula: ϕ = tan⁻¹((A1sin(ϕ1) + A2sin(ϕ2) + A3sin(ϕ3)) / (A1cos(ϕ1) + A2cos(ϕ2) + A3cos(ϕ3))). This gives the angle at which the resultant wave is shifted from the original waves.

3. Can the resultant amplitude be negative?

No, the resultant amplitude cannot be negative. It is always a positive value since it is the square root of the sum of squares of the individual amplitudes.

4. How does the phase difference between the individual waves affect the resultant amplitude?

The phase difference between the individual waves affects the resultant amplitude by causing constructive or destructive interference. If the phase differences are all the same, the waves will constructively interfere and the resultant amplitude will be larger. If the phase differences are different, the waves will destructively interfere and the resultant amplitude will be smaller.

5. Is the resultant amplitude affected by the wavelength of the individual waves?

Yes, the resultant amplitude is affected by the wavelength of the individual waves. This is because the phase difference between waves can be affected by the wavelength, which in turn affects the interference and the resultant amplitude.

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