How to Find Total Power/Intensity of a Light Beam Using Cylindrical Coordinates?

[shaf777]

Homework Statement

I have an equation to solve. See the attach pichture.
I(r,φ,λ) = I₀.sin(2πr/R).sin²(φ).(1/λ)
where
I₀ , R are constants
r,φ are cylindrical coordinates
0≤r≤R/2 : the cut is laterally limited
λ_a ≤ λ ≤ λ_b : the beam contains a limited frequency spektrum
where the shortcut for integration:
∫sin²(ax) dx = x/2 - 1/4a .sin(2ax)
∫x. sin(ax) dx = (sin(ax)/a²) - (x.cos(ax)/a)

The qustions are to find the total power/Intensity of the light beam, that was produced by λ.

Homework Equations

The Attempt at a Solution

I want to integrate the above equation to get the power or intensity of the beam, but I am stuck.
∫I(r, φ, λ) dλ = ∫ I₀ sin(2πr/R).sin²(φ).(1/λ) dλ
= sin(2πr/R).sin²(φ) ∫ (1/λ) dλ
= sin(2πr/R).sin²(φ) . ln|λ| + C

Is my answer correct? Because I did not even use the given Integration shortcut.

 

[BvU]

Hello shaf,

You integrate over $\lambda$. Is that really what's being asked ? Perhaps you can post the full problem statement ?

 

[shaf777]

Hello shaf,

You integrate over $\lambda$. Is that really what's being asked ? Perhaps you can post the full problem statement ?

Hello BvU,
actually this question is in German language (which gives it harder to understand) and I am not quite sure if the translation gives the right meaning or not. But if you can try to understand it a bit that would be helpful.

The question are:
a) Wie groß ist die gesamte Leistung diesen Strahl, die von allen λ erzeugt wird.

b) Wie groß ist die Leistung des Strahls, die von den λ in Intervall [λ₁, λ₂] mit λ₁, λ₂ ∈ λ_a, λ_b] in einem Kreis Segment zwischen der Radien ₁ und r₂ (r₁, r₂) ∈ [0, R] mit dem Azimutwinkel zwischen φ₁ und φ₂ ∈ [0, 2π] erzeugt wird.
Hier reicht es eine Brechungsformel aufzustellen. Die analytischen Ausdrücke müssen nicht weiter berechnet bzw. vereinfacht werden.

c) Wie musste die λ-Abhängigkeit in spektralen Intensitätsverteilung abgeändert werden, damit die spektrale Intensitätsverteilung, wenn man Sie als Funktion der Frequenz aufträgt eine Konstante bzgl. der Frequenz ergibt. Bergründen.

a) How big is the total intensity of this beam which is generated by all λ.
b) What is the intensity of the beam, emitted by the λ in interval [λ ₁ , λ ₂ ] with λ ₁ , λ ₂ ∈ λ _a , λ _b ] in a circle segment between the radii ₁ and r ₂ ] (r ₁ , r ₂ ) ∈ [0, R] with the azimuth angle between φ ₁ and φ ₂ ∈ [0, 2π] is generated.
Here it is enough to set up a formula of refraction. The analytical expressions need not be further calculated or simplified.
c) How did the λ-dependence in the spectral intensity distribution need to be changed so that the spectral intensity distribution, when plotted as a function of the frequency, gives a constant with respect to the frequency. Explain.

 

[BvU]

Question is one thing, problem statemenet is a little bit more

My dictionary says Leistung = Power. What is the relationship between your $I$ and power ? (From the hint I'd say I is power too... ?)
For that matter, I have no idea what $r$ and $\phi$ are exactly (your picture didn't make it).
But you are obviously expected to integrate over $r$ and $\phi$ too... "gesamte Leistung "

Brechungsformel

Berechnungsformel ?