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A uniform magnetic field B=B j^ exists in a space . A particle of mass m and charge q is projected towards negative x-axis with a speed v from the point (d,0,0). The maximum value v for which particle does not hit y-z plane?
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A uniform magnetic field B=B j^ exists in a space . A particle of mass...
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A uniform magnetic field B=B j^ exists in a space . A particle of mass...
Solution: Projectile motion in uniform magnetic field
Introduction
We are given a scenario of a charged particle moving in a uniform magnetic field and we are asked to find the maximum velocity at which the particle will not hit the y-z plane.
Theory
The motion of a charged particle in a uniform magnetic field is governed by the Lorentz force equation:
F = q(v x B)
where F is the force experienced by the particle, v is the velocity of the particle, B is the magnetic field and q is the charge of the particle.
The Lorentz force causes the charged particle to move in a circular path perpendicular to the magnetic field.
Solution
We are given that the particle is projected towards the negative x-axis with a velocity v from the point (d,0,0). The initial velocity vector of the particle is:
v = -v i^
where i^ is the unit vector along the x-axis.
The force experienced by the particle due to the magnetic field is:
F = q(v x B) = qvB k^
where k^ is the unit vector along the z-axis.
The magnitude of the force is given by:
|F| = qvB
This force causes the particle to move in a circular path in the y-z plane.
The radius of the circular path is given by:
r = mv / (qB)
where m is the mass of the particle.
The particle will hit the y-z plane when the radius of the circular path becomes equal to d. Therefore, we have:
mv / (qB) = d
Solving for v, we get:
v = dqB / m
This is the maximum velocity at which the particle will not hit the y-z plane.
Conclusion
In this problem, we used the Lorentz force equation to find the maximum velocity at which a charged particle will not hit the y-z plane when projected towards the negative x-axis in a uniform magnetic field.
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A uniform magnetic field B=B j^ exists in a space . A particle of mass m and charge q is projected towards negative x-axis with a speed v from the point (d,0,0). The maximum value v for which particle does not hit y-z plane?
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