To solve the problem of two charged particles moving in a magnetic field, we need to analyze their motion separately, taking into account their charges, masses, and the angle at which they are projected. Let's break this down step by step.
Understanding the Motion of Charged Particles in a Magnetic Field
When a charged particle moves through a magnetic field, it experiences a magnetic force that acts perpendicular to both the velocity of the particle and the magnetic field. This force is given by the Lorentz force equation:
F = q(v × B)
where F is the magnetic force, q is the charge of the particle, v is its velocity, and B is the magnetic field.
Analyzing Each Particle's Motion
We have two particles:
- Particle 1: Charge = +Q, Mass = M
- Particle 2: Charge = -Q, Mass = 2M
Both particles are projected with the same initial velocity v at an angle α to the magnetic field B.
Determining the Radius of Circular Motion
The magnetic force causes the particles to move in circular paths. The radius of the circular motion can be derived from the centripetal force equation:
F = (mv²)/r
Setting the magnetic force equal to the centripetal force, we have:
qvB = (mv²)/r
From this, we can solve for the radius r:
r = (mv)/(qB)
Calculating the Radius for Each Particle
For Particle 1 (+Q, M):
r₁ = (Mv)/(QB)
For Particle 2 (-Q, 2M):
r₂ = (2Mv)/(2QB) = (Mv)/(QB)
Interestingly, both particles have the same radius of circular motion, which is r = (Mv)/(QB).
Time Period of Circular Motion
The time period T for one complete revolution can be calculated using:
T = (2πr)/v
Substituting the radius we found:
T₁ = (2π(Mv)/(QB))/v = (2πM)/(QB) for Particle 1
T₂ = (2π(Mv)/(QB))/v = (2πM)/(QB) for Particle 2
Both particles have the same time period, which means they will complete their circular paths in the same amount of time.
Finding the Distance from the Projection Point
Since both particles have the same radius and time period, they will meet again after one complete revolution. The distance from the point of projection to the point where they meet again is equal to the circumference of the circular path:
Distance = 2πr
Substituting the radius:
Distance = 2π(Mv)/(QB)
Final Result
Thus, the distance from the point of projection where the two particles will meet again for the first time is:
Distance = (2πMv)/(QB)}
This result shows how the interplay of charge, mass, and the magnetic field influences the motion of charged particles, leading them to meet at a specific distance after being projected. Understanding these principles is crucial in fields like electromagnetism and particle physics.
