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. Let's break this down step by step.
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 inside the solenoid.
- 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 through a magnetic field, it experiences a magnetic force given by:
F = q(v × B)
In this case, since the particle is projected perpendicular to the axis of the solenoid, the force will act as a centripetal force, causing the particle to move in a circular path. The magnitude of this force can be expressed as:
F = qvB
Centripetal Force Requirement
For the particle to move in a circular path without striking the solenoid, the magnetic force must equal the required centripetal force:
F_c = m(v²/r)
where r is the radius of the circular path. Setting the magnetic force equal to the centripetal force gives us:
qvB = m(v²/r)
Solving for Maximum Speed
We can rearrange this equation to find the maximum speed v:
qv(μ₀nI) = m(v²/r)
Rearranging further, we get:
q(μ₀nI) = mv/r
From this, we can isolate v:
v = (qμ₀nI r) / m
Conclusion
The maximum speed at which the particle can be projected without striking the solenoid is given by:
v_{max} = (qμ₀nIr) / m
This formula indicates that the maximum speed depends on the charge and mass of the particle, the current in the solenoid, the number of turns per unit length, and the radius of the solenoid. If any of these parameters change, the maximum allowable speed for the particle will also change accordingly.
