A proton and an electron have the same kinetic energy upon entering a region of constant magnetic field. What is the ratio of the radii of their circular path?
To find the ratio of the radii of their circular paths, we need to use some basic principles of physics.
First, we can start by considering the equation for the centripetal force acting on a charged particle moving in a magnetic field. This force can be derived from the Lorentz force equation:
F = q(v⃗ × B⃗)
where F is the force, q is the charge of the particle, v⃗ is the velocity of the particle, and B⃗ is the magnetic field.
From this equation, we can deduce that the magnitude of the force experienced by a charged particle moving perpendicular to a uniform magnetic field is given by:
F = qvB
where v is the magnitude of the particle's velocity and B is the magnitude of the magnetic field.
Since the force acting on the particles is centripetal in nature, we can equate it to the centripetal force:
mv²/r = qvB
where m is the mass of the particle and r is the radius of the circular path.
We are given that the kinetic energy of both the proton and the electron is the same, which is given by:
K.E. = 1/2 mv²
Since the kinetic energies are the same, we can equate them:
1/2 m_proton v_proton² = 1/2 m_electron v_electron²
Now, to find the ratio of the radii, we can rearrange the equation to solve for r_proton/r_electron:
r_proton/r_electron = (m_electron/m_proton) * (v_proton/v_electron) * (B_electron/B_proton)
The ratio of their masses (m_electron/m_proton) is approximately 1/1836, since the mass of an electron is much smaller than that of a proton.
The ratio of their velocities (v_proton/v_electron) can be calculated using their kinetic energies. Since the kinetic energies are equal:
1/2 m_proton v_proton² = 1/2 m_electron v_electron²
v_proton/v_electron = √(m_electron/m_proton)
Finally, since the forces acting on the particles are perpendicular to each other, their magnetic field strengths (B) will be the same. Therefore, the ratio of the radii reduces to:
r_proton/r_electron = (m_electron/m_proton) * (v_proton/v_electron)
Substituting the values, we have:
r_proton/r_electron = (1/1836) * (√(m_electron/m_proton))
Please note that this calculation assumes that both the proton and the electron are moving perpendicular to the magnetic field and that the magnetic field is uniform.