In a certain cyclotron a proton moves in a circle of radius 0.5m. The magnitude of the magnetic field
is 1.2T.
a) What is the oscillator frequency?
b) What is the final kinetic energy of the proton?
3.3
To calculate the oscillator frequency and the final kinetic energy of the proton in the given cyclotron, we need to use the following formulas:
a) Oscillator Frequency:
The oscillator frequency (f) can be calculated using the formula:
f = 1 / (2π) * (q * B) / (m)
Where:
- f is the oscillator frequency.
- q is the charge of the proton (1.6 x 10^-19 C).
- B is the magnitude of the magnetic field (1.2 T).
- m is the mass of the proton (1.67 x 10^-27 kg).
Let's plug in the values and calculate the oscillator frequency.
f = 1 / (2π) * (1.6 x 10^-19 C * 1.2 T) / (1.67 x 10^-27 kg)
b) Final Kinetic Energy:
The final kinetic energy (KE) of the proton can be calculated using the formula:
KE = (1/2) * m * v^2
Where:
- KE is the kinetic energy.
- m is the mass of the proton (1.67 x 10^-27 kg).
- v is the final velocity of the proton.
To find the final velocity (v), we can use the following formula:
v = r * ω
Where:
- v is the final velocity.
- r is the radius of the proton's circular motion (0.5 m).
- ω is the angular velocity.
The angular velocity (ω) can be calculated using the formula:
ω = 2π * f
Now, let's calculate the final kinetic energy.
KE = (1 / 2) * (1.67 x 10^-27 kg) * (0.5 m * 2π * f)^2
Now we have the formulas and can calculate the answers to both parts of the question.