An electron and a proton move in circular orbits in a plane perpendicular to a uniform magnetic field.
Find the ratio of the radii of their circular orbits when the electron and proton have
A. the same momentum
B. the same kinetic energy
To find the ratio of the radii of their circular orbits, we need to use the equation for the magnetic force experienced by a charged particle moving in a magnetic field. The equation is given by:
F = qvB*sin(θ)
where F is the magnetic force, q is the charge of the particle, v is the velocity of the particle, B is the magnetic field, and θ is the angle between the velocity vector and the magnetic field vector.
A. Same Momentum:
When the electron and proton have the same momentum, we can equate their momenta:
m_e * v_e = m_p * v_p
where m_e and m_p are the mass of the electron and proton, respectively, and v_e and v_p are their velocities.
Since both particles have the same magnitude of charge (|q_e| = |q_p| = e), we can set their magnetic forces equal to each other:
q_e * v_e * B = q_p * v_p * B
e * v_e * B = e * v_p * B
From this equation, we can see that the ratio of velocities, v_e / v_p, is equal to 1.
Considering the fact that the magnetic force is given by F = mv^2 / r, where m is the mass of the particle and r is the radius of the circular orbit, and the ratio of velocities is 1, the ratio of the radii can be given as:
r_e / r_p = (m_p * v_p^2) / (m_e * v_e^2)
Since the mass of the proton (m_p) is greater than the mass of the electron (m_e) and the ratio of velocities is 1, the ratio of the radii will be greater than 1, indicating that the proton's orbit will have a larger radius than the electron's orbit.
B. Same Kinetic Energy:
When the electron and proton have the same kinetic energy, we can equate their kinetic energies:
(1/2) m_e * v_e^2 = (1/2) m_p * v_p^2
From this equation, we can see that the ratio of the square of velocities, v_e^2 / v_p^2, is equal to (m_p / m_e).
Using the same logic as in part A, considering the relationship between magnetic force and radius, we can write:
r_e / r_p = (m_p * v_p^2) / (m_e * v_e^2)
Substituting the ratio of velocities obtained from the kinetic energy equation, we get:
r_e / r_p = (m_p * v_p^2) / (m_e * v_e^2) = (m_p / m_e)
Since the mass of the proton (m_p) is greater than the mass of the electron (m_e), the ratio of the radii will be greater than 1, indicating that the proton's orbit will have a larger radius than the electron's orbit.