Answer:
When a charged particle, like a proton, travels in a circular orbit, it undergoes acceleration due to the centripetal force required to keep it in that path. Accelerated charged particles emit electromagnetic radiation in the form of synchrotron radiation. The power radiated by a charged particle moving in a circular orbit can be calculated using the Larmor formula:
Power radiated (P) = (q^2 * a^2 * c) / (6 * π * ε₀),
Step-by-step explanation:
When a charged particle, like a proton, travels in a circular orbit, it undergoes acceleration due to the centripetal force required to keep it in that path. Accelerated charged particles emit electromagnetic radiation in the form of synchrotron radiation. The power radiated by a charged particle moving in a circular orbit can be calculated using the Larmor formula:
Power radiated (P) = (q^2 * a^2 * c) / (6 * π * ε₀),
where:
q = charge of the particle,
a = acceleration of the particle,
c = speed of light in a vacuum, and
ε₀ = vacuum permittivity.
The acceleration (a) of the proton can be calculated from its centripetal acceleration, given by a = v^2 / r, where:
v = velocity of the proton,
r = radius of the circular orbit.
Now, we need to calculate the velocity (v) of the proton. The kinetic energy (KE) of the proton can be related to its speed using the equation:
KE = (1/2) * m * v^2,
where:
m = mass of the proton.
Given the kinetic energy of the proton is 5.5 MeV, we need to convert this to joules (J) since the units must be consistent for the subsequent calculations. 1 MeV is equal to 1.60218 × 10^-13 J, so:
KE = 5.5 MeV * 1.60218 × 10^-13 J/MeV.
Now, we can find the velocity (v) of the proton using the kinetic energy:
5.5 MeV * 1.60218 × 10^-13 J/MeV = (1/2) * m * v^2.
Let's rearrange the equation to solve for v:
v^2 = (2 * 5.5 MeV * 1.60218 × 10^-13 J/MeV) / m.
Now, we'll divide the kinetic energy by the mass of the proton. The mass of a proton is approximately 1.67262192 × 10^-27 kg.
v^2 = (2 * 5.5 MeV * 1.60218 × 10^-13 J/MeV) / 1.67262192 × 10^-27 kg.
v^2 ≈ 2.066866 × 10^13 m^2/s^2.
Taking the square root of both sides:
v ≈ √(2.066866 × 10^13) m/s.
v ≈ 4.54297 × 10^6 m/s.
Now, we can calculate the centripetal acceleration (a) using:
a = v^2 / r.
a ≈ (4.54297 × 10^6 m/s)^2 / 0.530 m.
a ≈ 4.60895 × 10^13 m/s^2.
Next, we can calculate the power radiated (P) by the proton using the Larmor formula:
P = (q^2 * a^2 * c) / (6 * π * ε₀).
The charge of a proton (q) is approximately 1.60217663 × 10^-19 C.
Substituting the known values:
P = (1.60217663 × 10^-19 C)^2 * (4.60895 × 10^13 m/s^2)^2 * c / (6 * π * ε₀).
The speed of light in a vacuum (c) is approximately 299,792,458 m/s, and the vacuum permittivity (ε₀) is approximately 8.854187817 × 10^-12 C^2/(N·m^2).
P ≈ (2.5665 × 10^-38 C^2) * (2.1257 × 10^27 m^2/s^4) * 299,792,458 m/s / (6 * π * 8.854187817 × 10^-12 C^2/(N·m^2)).
P ≈ 7.55906 × 10^-15 W.
Now, to find the fraction of energy radiated per second, we need to divide the power radiated (P) by the kinetic energy (KE) of the proton:
Fraction radiated per second = P / KE.
Fraction radiated per second ≈ (7.55906 × 10^-15 W) / (5.5 MeV * 1.60218 × 10^-13 J/MeV).
Fraction radiated per second ≈ 0.0869.
So, the proton radiates approximately 8.69% of its energy per second as it travels in a circular orbit in the particle accelerator.