Calculate the best angle for maximum light dispersion through a medium

In summary, the speaker initially thought that the maximum angle would occur when the angle is closest to the critical angle for total internal reflection. They used the equation for the critical angle and differentiated it with respect to wavelength to find the change in critical angle. They then solved for the wavelength at which the derivative is 0 in order to calculate the maximum angle. However, someone else pointed out that they should be maximizing wavelength with respect to angle instead.
  • #1
Jan Berkhout
8
3
Homework Statement
If you have light moving from ##n_1## to ##n_2## you can get dispersion if ##n_1## is a function of wavelength.
What angle of incidence ##(θ_1)## will maximise the dispersion for the situation below where the light goes from a medium with ##n_1(λ)## to vacuum, ##(n_2 = 1)##?
Relevant Equations
$$\frac{d \sin ^{-1}(\text{ax})}{\text{dx}}=\frac{a}{\sqrt{1-(ax)^2}}$$ $$\frac {dy}{dx}=\frac{dy}{dz} \frac{dz}{dx}$$
I first thought that the angle would have to be maximum when it is closest to the critical angle for total internal reflection. From my lectures the equation for the critical angle is ##\theta _1>\ sin ^{-1} \left( \frac {n_2} {n_1} \right),## so as ##n_2 = 1##, we have ##\theta _1=\sin ^{-1}\left(\frac{1}{n_1(\lambda)}\right)##. I didn't really know what to do after that but from the equations given in the hint (relevant equations), I thought I'd have to differentiate with respect to ##\lambda##. This gives $$\frac {d θ_1} {d \lambda} = -\frac{\frac{d}{{d \lambda}} n_1(\lambda)}{n_1(\lambda) \sqrt{n_1(\lambda)^2-1}}.$$ So I have the change in the critical angle with respect to the change in wavelength, so my hunch is I have to set the derivative to 0 and solve for \lambda to find the wavelength for the maximum angle then I can calculate the angle? Is this right?
 
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  • #2
I worked hard and actually believe I have solved it! So this thread can be closed :)
 
  • #3
Jan Berkhout said:
This gives $$\frac {d θ_1} {d \lambda} = -\frac{\frac{d}{{d \lambda}} n_1(\lambda)}{n_1(\lambda) \sqrt{n_1(\lambda)^2-1}}.$$
Don't you want to maximise ##\lambda## wrt ##\theta_1##? For that you need $$\frac {d \lambda} {d θ_1}$$.
 

Related to Calculate the best angle for maximum light dispersion through a medium

1. What is the purpose of calculating the best angle for maximum light dispersion through a medium?

The purpose of this calculation is to determine the optimal angle at which light can pass through a medium with the least amount of obstruction or loss. This is important in various fields such as optics, photography, and material science.

2. How is the best angle for maximum light dispersion through a medium calculated?

The best angle is typically calculated using the laws of refraction, which describe how light bends when passing through different mediums. By knowing the refractive index of the medium and the angle of incidence, the angle of refraction can be calculated using Snell's law. The angle at which the refracted light is perpendicular to the surface of the medium is considered the optimal angle for maximum light dispersion.

3. What factors can affect the best angle for maximum light dispersion?

The refractive index of the medium, the angle of incidence, and the wavelength of light are the main factors that can affect the best angle for maximum light dispersion. Additionally, the surface properties and thickness of the medium can also play a role in determining the optimal angle.

4. Why is it important to consider the best angle for maximum light dispersion in experiments or applications?

In experiments or applications involving light passing through a medium, it is important to consider the best angle for maximum light dispersion in order to minimize any loss or distortion of the light. This can help improve the accuracy and precision of measurements, as well as ensure that the desired outcome is achieved.

5. Can the best angle for maximum light dispersion change for different mediums?

Yes, the best angle for maximum light dispersion can vary for different mediums due to differences in their refractive indices. The optimal angle for one medium may not be the same for another, so it is important to calculate it for each specific medium in order to achieve the best results.

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