An electron and a proton move in circular orbits in a plane perpendicular to a uniform magnetic field B.

Find the ratio of the radii of their circular orbits when the electron and the proton have the same momentum.

Find the ratio of the radii of their circular orbits when the electron and the proton have the same kinetic energy.

To find the ratio of the radii of the circular orbits for the electron and the proton, we can use a combination of the equations for centripetal force, magnetic force, and momentum.

1. Same Momentum:
The momentum of a particle is given by the equation:
p = mv,
where p is the momentum, m is the mass, and v is the velocity.

Since the electron and the proton both have the same momentum, we can equate their momenta:
me * ve = mp * vp,
where me is the mass of the electron, ve is the velocity of the electron, mp is the mass of the proton, and vp is the velocity of the proton.

For a charged particle moving in a uniform magnetic field, the magnetic force acting on the particle is given by:
F = qvB,
where F is the magnetic force, q is the charge, v is the velocity, and B is the magnetic field strength.

Since the particles are moving in circular orbits, the magnetic force provides the centripetal force:
F = mv^2 / r,
where r is the radius of the circular orbit.

For the electron:
me * ve^2 / re = e * ve * B,
where e is the elementary charge.

For the proton:
mp * vp^2 / rp = e * vp * B.

Now, divide the equation corresponding to the electron by the equation corresponding to the proton:
(me / mp) * (ve^2 / vp^2) * (rp / re) = (ve / vp).

Notice that the ratio (me / mp) is small, approximately 1/1836. Also, the ratio (ve / vp) is approximately 1 because they have the same momentum. Thus, we can ignore these terms in the equation.

Simplifying the equation gives:
(rp / re) = (vp^2 / ve^2).

Therefore, the ratio of the radii of their circular orbits when the electron and the proton have the same momentum is equal to the ratio of the squares of their velocities.

2. Same Kinetic Energy:
The kinetic energy of a particle is given by the equation:
KE = (1/2)mv^2.

Since the electron and the proton have the same kinetic energy, we can equate their kinetic energies:
(1/2)me * ve^2 = (1/2)mp * vp^2.

Simplifying the equation gives:
me * ve^2 = mp * vp^2.

Using the same derivation as before, the ratio of the radii of their circular orbits when they have the same kinetic energy is equal to the ratio of the squares of their velocities.

In summary:
1. Same Momentum: (rp / re) = (vp^2 / ve^2);
2. Same Kinetic Energy: (rp / re) = (vp^2 / ve^2).