# Interference Problem: Finding Maximum Radiation Intensity

• Saitama
In summary, the problem at hand involves a system with two coherent point sources located in a certain plane with dipole moments oriented at right angles to that plane. The sources are separated by a distance d and have a radiation wavelength of ##\lambda##. The problem asks to find the angles ##\theta## at which the radiation intensity is maximum, taking into account that the oscillations of the second source lag in phase behind the oscillations of the first source by ##\phi##. This lag in phase is in time, not distance, and can be incorporated into the phase difference between the two sources' waves.
Saitama

## Homework Statement

A system illustrated in the 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.

## The Attempt at a Solution

Let the radiations from the sources ##S_1## and ##S_2## interfere at a point P. As the radiation from ##S_1## is already ahead by the given phase, I have
$$S_2P=S_1 P+d\cos\theta-\frac{\phi \lambda}{2\pi}$$
For constructive interference,
$$S_2P-S_1P=n\lambda$$
But solving the above equations doesn't give the right answer.

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Pranav-Arora said:

## Homework Statement

A system illustrated in the 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.

## The Attempt at a Solution

Let the radiations from the sources ##S_1## and ##S_2## interfere at a point P. As the radiation from ##S_1## is already ahead by the given phase, I have
$$S_2P=S_1 P+d\cos\theta-\frac{\phi \lambda}{2\pi}$$

If you mean that S2P and S1P are distances of the point P from the sources, S2P - S1P≈dcosθ (when d<<S1P and d<<S2P). The phase difference between the rays arriving at P is (2π/λS2P-Φ -2π/λS1P), the effective path difference is S2P-S1P-Φλ/(2π) and that must be equal to integer times lambda.

ehild

1 person
Hi ehild! :)

ehild said:
If you mean that S2P and S1P are distances of the point P from the sources, S2P - S1P≈dcosθ (when d<<S1P and d<<S2P). The phase difference between the rays arriving at P is (2π/λS2P-Φ -2π/λS1P), the effective path difference is S2P-S1P-Φλ/(2π) and that must be equal to integer times lambda.

ehild

Your method makes sense to me but solving it further doesn't seem to give the correct answer. Since ##S_2P-S_1P=d\cos\theta##, for constructive interference, I get:
$$d\cos\theta-\frac{\phi \lambda}{2\pi}=n\lambda \Rightarrow \cos\theta=\left(n+\frac{\phi}{2\pi}\right)\frac{\lambda}{d}$$
But the given answer is:
$$\cos\theta=\left(n-\frac{\phi}{2\pi}\right)\frac{\lambda}{d}$$

It appears to me that the problem is worded such that dipole 2 lags dipole 1 in time. So, the phase lag in wave 2 is a phase lag in time, not distance.

TSny said:
It appears to me that the problem is worded such that dipole 2 lags dipole 1 in time. So, the phase lag in wave 2 is a phase lag in time, not distance.

You are right, the phase of the light from source 2 is (2pi/λ)s2-(ωt-Φ) which results in a positive phase difference in distance. ehild

TSny said:
It appears to me that the problem is worded such that dipole 2 lags dipole 1 in time. So, the phase lag in wave 2 is a phase lag in time, not distance.

Sorry if this is a stupid question but what is the difference between the two?

Read my previous post. -Φ is associated with the time term in the phase of the wave. You get "+Φ" added to the space part.

$$d\cos\theta+\frac{\phi \lambda}{2\pi}=n\lambda$$

ehild

ehild said:
Read my previous post. -Φ is associated with the time term in the phase of the wave. You get "+Φ" added to the space part. $$d\cos\theta+\frac{\phi \lambda}{2\pi}=n\lambda$$ehild

I still don't seem to understand the difference between "phase lag in time" and "phase lag in distance". :(

Working backwards from the answer, the phase difference must be
$$\frac{2\pi}{\lambda}S_2P+\phi-\frac{2\pi}{\lambda}S_1P$$
But I don't get why we add ##\phi##. What is the difference in answer if it is mentioned that "phase lag is in distance"?

Last edited:
A traveling wave is of the form E=Eosin(ωt-kr). If the oscillation of S2 lags behind S1, the time lag appears in the ωt term : it becomes ω(t-Δt). The phase changes by -ωΔt=-Φ, but you can incorporate that Φ into the -kr term, with positive sign.

ehild

ehild said:
A traveling wave is of the form E=Eosin(ωt-kr). If the oscillation of S2 lags behind S1, the time lag appears in the ωt term : it becomes ω(t-Δt). The phase changes by -ωΔt=-Φ, but you can incorporate that Φ into the -kr term, with positive sign.

ehild

Sorry if this is going to sound stupid but in this case, would it be correct to say that oscillation of S2 lags by ##\phi## in time but is ahead by ##\phi## in distance?

Also, how do you conclude from the problem statement that the phase lag is in time not in distance?

Pranav-Arora said:
Sorry if this is going to sound stupid but in this case, would it be correct to say that oscillation of S2 lags by ##\phi## in time but is ahead by ##\phi## in distance?

No, I do not think so. Lag means something that happens later. There is phase difference because of time lag, and because of longer distance travelled.

Pranav-Arora said:
Also, how do you conclude from the problem statement that the phase lag is in time not in distance?

That was the problem statement:
Taking into account that the oscillations of source 2 lag in phase behind the oscillations of source 1 by ϕ...

The sources are oscillating dipoles. Their motion is not in phase.
The interference happens between the waves. Their phase difference is due to the time lag and the path difference.
ehild

## 1. What is an interference problem in the context of finding maximum radiation intensity?

An interference problem occurs when two or more waves overlap and interfere with each other, resulting in either constructive or destructive interference. In the context of finding maximum radiation intensity, this refers to the interference of multiple sources of radiation that can affect the overall intensity of the radiation at a specific location.

## 2. How does interference affect the maximum radiation intensity?

The interference of multiple sources of radiation can either add up (constructive interference) or cancel out (destructive interference) at a specific location. This means that interference can either increase or decrease the maximum radiation intensity at that location.

## 3. What factors can influence the interference problem in finding maximum radiation intensity?

The factors that can influence the interference problem include the number of sources of radiation, their relative distances from each other and from the location of interest, the frequency and wavelength of the radiation, and the direction of the waves.

## 4. How can the maximum radiation intensity be calculated in an interference problem?

The maximum radiation intensity can be calculated by taking into account the characteristics of the individual sources of radiation, such as their amplitudes and phases, and then applying the principles of wave interference to determine the overall intensity at a specific location.

## 5. What are some real-world applications of the interference problem in finding maximum radiation intensity?

Interference problems are commonly encountered in fields such as radio and telecommunications, where multiple sources of electromagnetic radiation can interfere with each other and affect the overall signal strength. Other applications include optics, acoustics, and quantum mechanics, where the principles of interference are used to study and manipulate waves.

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