The magnetic force on a particle carrying a charge q and moving with a velocity \(\vec{v}\) in a magnetic field of induction \(\vec{B}\) is \(\vec{F_m}\)= q\(\vec{v}×\vec{B}\). At every instant, \(\vec{F_m}\) is perpendicular to the linear velocity \(\vec{v}\) and \(\vec{B}\).
Therefore, a non—zero magnetic force may change the direction of the velocity and the dot product .\(\vec{F_m}.\vec{v}\) = \(q(\vec{v}×\vec{B}).\vec{v}\) = 0.
But \(\vec{F_m}.\vec{v}\) is the power, i.e., the time rate of doing work. Hence, the work done by the magnetic force in every short displacement of the particle is zero.
The work done by a force produces a change in kinetic energy. Zero work means no change in kinetic energy. Thus, although the magnetic force changes the direction of the velocity \(\vec{v}\), it cannot change the linear speed and the kinetic energy of the particle.
