Computing Cherenkov angle when only knowing n?

[1a2]

Homework Statement

If all we know is that the media is 1.001, how can we compute the Cherenkov angle and threshold momentum of electrons,muons, pions, and kaons?

Homework Equations

I know that cos(theta_c) = 1/beta*n, where beta = v_p/c and n=c/v_em, where v_p is the particle velocity and v_em is the velocity of the emitted wave.

The Attempt at a Solution

The problem doesn't us what v_p and v_em is. It only tells us the value of n. How are we supposed to find the Cherenkov angle?

 

[mfb]

Can you write down the exact problem statement?

The problem might ask for the maximal cherenkov angle.

You know v_em as you know n and c.

 

[1a2]

Here is the exact problem statement:

Calculate the Cherenkov angle and the threshold momentum for media with n=1.001, 1.01, 1.03, 1.07, 1.12, 1.33, 1.50 for electrons, muons, pions, kaons, and protons

 

[mfb]

Hmm... then I guess it asks for the maximal Cherenkov angle.
At least threshold momentum is clear.

 

[1a2]

so how am I supposed to solve for the Cherenkov angle then, if we can't obtain v_p and beta? Should I ask the instructor for a hint or more clarifiation?

 

[mfb]

At which speed do you get the maximal angle?

Asking the instructor for clarification does not harm I guess, but the question above can be answered with the given information.

 

[1a2]

the max angle is 90 degrees, right? That occurs when cos(theta_c) = 0 ⇒ v_em = v_p? I don't understand how finding the maximal angle is related to the Cherenkov angle

 

[mfb]

How can you get cos(theta_c)=1/(beta*n)=0? How can 1 divided by something be zero?

 

[1a2]

How can you get cos(theta_c)=1/(beta*n)=0? How can 1 divided by something be zero?

so the max angle occurs when beta→∞. But beta = v_p/c, so v_p→∞

So we should just set v_p→∞ for each of the values for n? The problem asks for the angles, not v_p values

 

[mfb]

The speed cannot go to infinity, there is a very hard limit to speeds in the universe.

 

[1a2]

so when v_p→c? But then that gives beta→1 and thus cos(θ_c)→1/(1*n)

so for n=1.001, 1.01, 1.03, 1.07, 1.12, 1.33, 1.50, θ_c = cos^-1(1/n)?

 

[mfb]

Right.