Periodic Time for Alpha Particle in a Uniform Magnetic Field

In order to determine the periodic time for an alpha particle in a uniform magnetic field, we need to consider the relationship between the charge, mass, velocity, and magnetic field.

Given Information:
- Speed of the proton = Speed of the alpha particle
- Proton takes 20 microseconds to make 5 revolutions

Understanding the Situation:
When a charged particle enters a uniform magnetic field perpendicular to its direction of motion, it experiences a force called the magnetic force. This force acts as a centripetal force, causing the particle to move in a circular path. The frequency of the circular motion determines the periodic time of the particle.

Key Concepts:
1. The centripetal force acting on a charged particle moving in a magnetic field is given by the equation F = qvB, where F is the force, q is the charge of the particle, v is its velocity, and B is the magnetic field strength.
2. The centripetal force is also given by the equation F = (mv^2)/r, where m is the mass of the particle and r is the radius of the circular path.
3. Equating these two equations, we get qvB = (mv^2)/r.
4. The velocity of a charged particle moving in a circular path is given by the equation v = 2πr/T, where T is the periodic time.
5. Substituting this velocity equation into the equation from step 3, we get q(2πr/T)B = (m(2πr/T)^2)/r.
6. Simplifying the equation, we find qB = (4π^2mr)/T^2.

Calculating the Periodic Time:
To find the periodic time for the alpha particle, we can use the equation derived in step 6.

1. Let's assume the charge of the alpha particle is qα and the charge of the proton is qp.
2. Since both particles have the same speed, their velocities (vα and vp) are equal.
3. The ratio of their charges is qα/qp = 2.
4. The ratio of their masses is mα/mp = 4, where mα is the mass of the alpha particle and mp is the mass of the proton.
5. We know that the proton takes 20 microseconds to make 5 revolutions, which means Tp = 20 microseconds / 5 = 4 microseconds per revolution.
6. Substituting the given values and ratios into the equation from step 6, we get qαB = (4π^2mα)/Tα^2.
7. Since qα/qp = 2 and mα/mp = 4, the equation becomes 2qpB = (4π^2mp)/Tα^2.
8. By rearranging the equation, we can solve for Tα: Tα^2 = (4π^2mp)/(2qpB).
9. Taking the square root of both sides, we find Tα = 2π√(mp/(qpB)).

Conclusion:
The periodic time for the alpha particle in a uniform magnetic field can be calculated using the equation Tα = 2π√(mp/(qpB)),