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As a magnetic field of intensity, $\mathbf{B}$ applies a force of $\mathbf{F}= q\mathbf{v} \times \mathbf{B}$ on a charge $q$ moving with velocity $\mathbf{v}$, this force $\mathbf{F}$ should be accelerating the charge, so it's its kinetic energy should increase. But that doesn't seem to happen, why? I know that the work done is zero because the force acts perpendicularly to the direction of the velocity vector, but there acts a force, and it should be accelerating the charge and it should be increasing its speed in some direction whatsoever, but that force seems only to change the charged particle's direction, why is that so?

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Acceleration is a change in velocity. Since velocity is a vector quantity there are two ways in which it can change. The magnitude (speed) can change, or the direction can change (or both at the same time).

For the magnetic force it always acts perpendicular to the velocity. Hence this force only produces a change in the direction of the velocity. This is still an acceleration, since the velocity vector is changing.

Therefore, to address misconceptions I see in your question:

  1. Acceleration does not mean change in kinetic energy (although a change in kinetic energy must be accompanied by an acceleration).
  2. Acceleration does not mean change in speed (although a change in speed must be accompanied by an acceleration).
  3. The magnetic force, just like all forces, do produce accelerations (assuming there are not other forces that counteract this force).

why do perpendicular forces only contribute to change in direction but not for the change in speed?

The velocity changes in the direction of the force (i.e. the direction of the acceleration). If the force is always perpendicular to the velocity, then you are not adding anything to the "length" (magnitude) of the vector. Think about your usual examples of centripetal forces and circular motion (like a car going around a curve, or spinning an object around on a string). The centripetal force only causes the direction of the velocity to change. In order to change the speed you need a force along the velocity.

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  • $\begingroup$ That almost solved my problem. But, why do perpendicular forces only contribute to change in direction but not for the change in speed? Try giving some Intuition (by using some physical example), rather than mathematical equations, so that I can understand better. $\endgroup$ – Mohammad Vajid Jul 9 at 16:27
  • $\begingroup$ @MohammadVajid Done $\endgroup$ – Aaron Stevens Jul 9 at 16:31
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    $\begingroup$ Interesting footnote: The magnetic force in itself would not cause the speed of the charged particle to change but due to the fact that it accelerates the charge (i.e., changes direction of the velocity of the charge), the charge would radiate and the charge would experience an Abraham-Lorentz force which will change its speed as well. $\endgroup$ – Dvij Mankad Jul 9 at 16:42
  • $\begingroup$ @FeynmansOutforGrumpyCat Yes, this is a good point. Thanks $\endgroup$ – Aaron Stevens Jul 9 at 16:45
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    $\begingroup$ @MohammadVajid The object does "speed up" in the perpendicular direction. If the force stayed in the same direction then certainly at the next instant the speed would be increasing. For example, in projectile motion, when the object reaches its maximum height the force is perpendicular to the velocity. Then the object starts moving downwards and its speed increases. But in the magnetism case the force is constantly changing so that it is always perpendicular to the velocity vector. Hence the overall speed never changes. $\endgroup$ – Aaron Stevens Jul 9 at 16:54
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... there acts a force, ... but that force seems only to change the charged particle's direction, why is that so?

What we observe in the interaction between a moving electron and an external magnetic field:

  • If the direction of the electron movement and the direction of the magnetic field are parallel, nothing happens, the electron moves straight ahead. Except for one small peculiarity: The electron has its own magnetic dipole and this dipole is aligned to the external magnetic field.
  • If the movement of the electron is nonparallel to the magnetic field and at minimum four things happens to the electron:
    1. The electrons magnetic moment is aligned by the magnetic field.
    2. The electron is deflected perpendicular to the plane between the motion vector of the electron and the direction of the magnetic field.
    3. The electron emits photons.
    4. The emission of photons reduces the kinetic energy of the electron (The emission of photons reduces the kinetic energy of the electron (and the electron moves in a spiral path until it exhausts all the kinetic energy and comes to a standstill in the centre of the spiral).

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What has not been considered so far is the sequence of the these events:

  • Does the alignment of the magnetic dipole led to the emission of photons?
  • Does the photon emission led to a tiny deflection of the electron and to a miss-alignment of the electrons magnetic dipole again?

This seems to be an interesting area of research.

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