Why a particle moving parallel to magnetic field has no force acting on it?

Question

Answer

Because the cross product of the velocity with magnetic field vanishes when they're both in the same direction.

Answer

Newton's first law of motion states that if an object experiences no net force, then its velocity is constant. ... If a charged particle's velocity is completely parallel to the magnetic field, the magnetic field will exert no force on the particle and thus the velocity will remain constant.

Answer

Introduction:
When a charged particle moves parallel to a magnetic field, it experiences no force. This phenomenon is known as the magnetic force on a moving charge. To understand why there is no force acting on a particle moving parallel to a magnetic field, let's delve into the physics behind it.

Explanation:
When a charged particle moves through a magnetic field, it experiences a force known as the Lorentz force. The Lorentz force can be represented by the equation:

F = q(v x B),

where F is the force experienced by the particle, q is the charge of the particle, v is its velocity, and B is the magnetic field vector. The vector product (v x B) indicates that the force acts perpendicular to both the velocity and magnetic field.

Case 1: Particle moving perpendicular to magnetic field:
When a charged particle moves perpendicular to a magnetic field, the force experienced by the particle is maximum. This force alters the particle's direction, causing it to move in a circular path. The radius of this circular path can be calculated using the equation:

F = mv²/r,

where m is the mass of the particle, v is its velocity, and r is the radius of the circular path. By equating these two equations, we find that:

qvB = mv²/r.

Simplifying this equation, we get:

r = mv/qB.

Case 2: Particle moving parallel to magnetic field:
When a charged particle moves parallel to a magnetic field, the force experienced by the particle is zero. This can be explained by considering the equation for the Lorentz force. Since the velocity (v) and the magnetic field (B) are parallel to each other, the cross-product (v x B) becomes zero. Therefore, the Lorentz force (F) also becomes zero.

Conclusion:
In conclusion, a charged particle moving parallel to a magnetic field experiences no force because the vector product of its velocity and magnetic field is zero. This phenomenon is consistent with the principles of electromagnetism and can be explained using the Lorentz force equation. Understanding the behavior of particles in magnetic fields is crucial in various scientific and technological applications, such as particle accelerators and magnetic resonance imaging (MRI) machines.

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