To determine the maximum speed at which a charged particle can be projected from the axis of a tightly wound solenoid without striking it, we need to analyze the forces acting on the particle and the magnetic field produced by the solenoid.
Understanding the Magnetic Field of the Solenoid
A long solenoid generates a uniform magnetic field inside it, given by the formula:
B = μ₀nI
where:
- B is the magnetic field strength.
- μ₀ is the permeability of free space (approximately 4π × 10-7 T·m/A).
- n is the number of turns per unit length of the solenoid.
- I is the current flowing through the solenoid.
Force on the Charged Particle
When a charged particle with charge Q moves in a magnetic field, it experiences a magnetic force given by:
F = QvB
where v is the velocity of the particle. This force acts perpendicular to both the velocity of the particle and the magnetic field, causing the particle to undergo circular motion if it enters the magnetic field region.
Condition for Not Striking the Solenoid
For the particle to avoid striking the solenoid, it must not enter the region where the magnetic field is present. This means that the particle should have enough initial velocity to travel a distance equal to the radius r of the solenoid before it starts to feel the magnetic force. The time t taken to travel this distance can be expressed as:
t = r/v
Magnetic Force and Circular Motion
If the particle were to enter the magnetic field, it would start moving in a circular path due to the magnetic force. The radius of this circular path R is given by:
R = (Mv)/(QB)
To ensure that the particle does not strike the solenoid, the distance it travels in the time t must be greater than or equal to the radius of the solenoid:
r ≤ R
Deriving the Maximum Speed
Substituting for R gives:
r ≤ (Mv)/(QB)
Rearranging this inequality to solve for v gives:
v ≤ (QBr)/M
Now, substituting the expression for B into this equation:
v ≤ (Q(μ₀nI)r)/M
Final Expression for Maximum Speed
This final expression provides the maximum speed at which the particle can be projected without striking the solenoid:
v_max = (Qμ₀nIr)/M
In summary, the maximum speed of the charged particle is directly proportional to the charge of the particle, the current in the solenoid, and the number of turns per unit length, while inversely proportional to the mass of the particle. This relationship highlights the balance between the magnetic force acting on the particle and its inertia, ensuring it can safely travel without entering the solenoid's magnetic field.
