Using a vacuum chamber of diameter 79.0 cm you want to create a cyclotron that accelerates protons to 21.0 % of the speed of light. What strength of magnetic field is required in order for this to work?Magnitude:
Question
Using a vacuum chamber of diameter 79.0 cm you want to create a cyclotron that accelerates protons to 21.0 % of the speed of light. What strength of magnetic field is required in order for this to work?Magnitude:
Solution
To solve this problem, we need to use the formula for the cyclotron frequency, which is given by:
f = qB / (2πm)
where:
- f is the frequency of the cyclotron,
- q is the charge of the proton,
- B is the magnetic field strength, and
- m is the mass of the proton.
We also know that the speed of the proton is given by:
v = 2πrf
where:
- v is the speed of the proton, and
- r is the radius of the cyclotron.
We can rearrange this equation to solve for f:
f = v / (2πr)
Substituting this into the cyclotron frequency equation gives:
v / (2πr) = qB / (2πm)
Rearranging this equation to solve for B gives:
B = mv / (qr)
We can now substitute the given values into this equation to find the magnetic field strength. The charge of a proton is 1.602 x 10^-19 C, the mass of a proton is 1.673 x 10^-27 kg, the speed of the proton is 0.21 times the speed of light (which is approximately 3.00 x 10^8 m/s), and the radius of the cyclotron is half the diameter, or 0.395 m.
B = (1.673 x 10^-27 kg * 0.21 * 3.00 x 10^8 m/s) / (1.602 x 10^-19 C * 0.395 m)
Solving this equation gives the magnetic field strength required for the cyclotron to work.
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