Chrenkov radiation cone angle vs. the speed of the particle

[help I have 12 hours]

Homework Statement

Electromagnetic radiation (known as Cherenkov radiation) is emitted when a charged particle moves through a medium faster then the local speed of light. It should be stressed that the particle is never going faster then the speed of light in vacuum (or c), just faster than the speed of light in the material (which is always less than c).

When a charged particle passes straight through a medium faster than the local speed of light, it will emit Cherenkov radiation in a cone. Let's see how the cone angle is correlated to the speed of the particle.

Ring-imaging Cherenkov detectors are devices that can accurately measure the velocity of charged particles as they pass through them. They are very useful as subdetectors in large particle detector systems. Let us look at what decisions need to go into designing a Cherenkov detector.


Suppose our detector used aerogel (n=1.03) for the Cherenkov material and the photomultiplier tube array had a resolution of Δθ=1.2degrees. This means, for instance, that the detector can distinguish between Cherenkov light emitted at an angle of 15 degrees and that emitted at 13.8 degrees but can't tell the difference between Cherenkov light emitted at 15 degrees and that emitted at 13.9 degrees. What is the highest velocity vmax at which a charged particle can be accurately measured to be below the speed of light in vacuum (c
)?
Express your answer as a multiple of c
to three significant figures.
vmax= _________________________(c)

Relevant Equations

cos(theta)=c/vn

where theta is the angle at which the cherenkov radiation is emitted.
n is index of refraction
v is velocity of the particle
c speed of light

cos(theta)=c/vn
cos(1.2 degrees)=((3*10^8)/v(1.03))
v=291262135.9

v/c=.971

so the particle is moving at (.971)c

But mastering physics tells me this is wrong, and says that I'm not account for the angle.
I am really stumped.

 

[haruspex]

Homework Statement:: Electromagnetic radiation (known as Cherenkov radiation) is emitted when a charged particle moves through a medium faster then the local speed of light. It should be stressed that the particle is never going faster then the speed of light in vacuum (or c), just faster than the speed of light in the material (which is always less than c).

When a charged particle passes straight through a medium faster than the local speed of light, it will emit Cherenkov radiation in a cone. Let's see how the cone angle is correlated to the speed of the particle.

Ring-imaging Cherenkov detectors are devices that can accurately measure the velocity of charged particles as they pass through them. They are very useful as subdetectors in large particle detector systems. Let us look at what decisions need to go into designing a Cherenkov detector.Suppose our detector used aerogel (n=1.03) for the Cherenkov material and the photomultiplier tube array had a resolution of Δθ=1.2degrees. This means, for instance, that the detector can distinguish between Cherenkov light emitted at an angle of 15 degrees and that emitted at 13.8 degrees but can't tell the difference between Cherenkov light emitted at 15 degrees and that emitted at 13.9 degrees. What is the highest velocity vmax at which a charged particle can be accurately measured to be below the speed of light in vacuum (c
)?
Express your answer as a multiple of c
to three significant figures.
vmax= _________________________(c)
Relevant Equations:: cos(theta)=c/vn

where theta is the angle at which the cherenkov radiation is emitted.
n is index of refraction
v is velocity of the particle
c speed of light

cos(theta)=c/vn
cos(1.2 degrees)=((3*10^8)/v(1.03))
v=291262135.9

v/c=.971

so the particle is moving at (.971)c

But mastering physics tells me this is wrong, and says that I'm not account for the angle.
I am really stumped.

You have found the velocity at which the angle of the cone would be 1.2 degrees. This is not what is asked.
You do not know the angle of the cone (yet), only the uncertainty in whatever cone is measured.

Consider two particles at the same point, one at the v_max to be found and one at, effectively, c. Draw the diagram showing both cones.
What is the angle of the speed c cone?
What do you know about the difference in the two cone angles?

 

[help I have 12 hours]

thank you, I got it.

 

[DifferentialGalois]

You have found the velocity at which the angle of the cone would be 1.2 degrees. This is not what is asked.
You do not know the angle of the cone (yet), only the uncertainty in whatever cone is measured.

Consider two particles at the same point, one at the v_max to be found and one at, effectively, c. Draw the diagram showing both cones.
What is the angle of the speed c cone?
What do you know about the difference in the two cone angles?

I am yet to have got it. I would like some mathematical reasoning if possible, i.e. the solution.

 

[haruspex]

I am yet to have got it. I would like some mathematical reasoning if possible, i.e. the solution.

If the particle's speed were extremely close to c, what would the cone angle be?
What is the largest actual cone angle that the detector can distinguish from that?

 

[DifferentialGalois]

If the particle's speed were extremely close to c, what would the cone angle be?
What is the largest actual cone angle that the detector can distinguish from that?

can u just give me the complete solution?

 

[haruspex]

can u just give me the complete solution?

It is against the rules of this forum to provide solutions. We point out mistakes, provide hints and suggest approaches.
Please try to answer my questions.

 

[DifferentialGalois]

It is against the rules of this forum to provide solutions. We point out mistakes, provide hints and suggest approaches.
Please try to answer my questions

would DM'ing me the solution help?

 

[haruspex]

would DM'ing me the solution help?

Of course not.

 

[DifferentialGalois]

If the particle's speed were extremely close to c, what would the cone angle be?
What is the largest actual cone angle that the detector can distinguish from that?

ok, cone angle is 0. largest cone angle is 90. amirite?

 

[haruspex]

ok, cone angle is 0. largest cone angle is 90. amirite?

No. For my first question use the "Relevant Equation" in post #1.

 

[DifferentialGalois]

No. For my first question use the "Relevant Equation" in post #1.

me lly no idea.

 

[haruspex]

me lly no idea.

The equation is $\cos(\theta)=\frac c{vn}$.
What is the value of n in the question? So what is the maximum $\theta$?

 

[DifferentialGalois]

The equation is $\cos(\theta)=\frac c{vn}$.
What is the value of n in the question? So what is the maximum $\theta$?

oh sorry i had no idea what i was talking about:
Here is a related problem that I wish to solve:

If a charged particle moves faster than the speed of light in a dielectric material along a straight line, then it will emit Cherenkov radiation in a conical shape. Consider a charged particle which moves along a circle at constant superluminal speed. Determine the shape of the wave front in the plane of the circle. Which known curves can describe the shape of the wave front in different regions of the plane of the circle?

Plot the wave front at different values of the ratio v/c, where v is the speed of the particle, and c is the speed of light in the material.

 

[haruspex]

oh sorry i had no idea what i was talking about:
Here is a related problem that I wish to solve:

If a charged particle moves faster than the speed of light in a dielectric material along a straight line, then it will emit Cherenkov radiation in a conical shape. Consider a charged particle which moves along a circle at constant superluminal speed. Determine the shape of the wave front in the plane of the circle. Which known curves can describe the shape of the wave front in different regions of the plane of the circle?

Plot the wave front at different values of the ratio v/c, where v is the speed of the particle, and c is the speed of light in the material.

ok, so, per forum rules, please post your attempt.

 

[DifferentialGalois]

ok, so, per forum rules, please post your attempt.

i drew a rough diagram and thats it

 

[haruspex]

i drew a rough diagram and thats it

So post it.

 

[berkeman]

The current poster has left the building for various reasons. Thanks for trying to help them despite their refusal to show work.