Question
Explain why the magnetic force on a charged particle cannot change the linear speed and the kinetic energy of the particle.###One implication of the Lorentz force law is that magnetic force does no work. Justify.
✓
Answer
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 $\vec{F}_{ m }=q \vec{v} \times \vec{B}$. At every instant, $\vec{F}_{ m }$ is perpendicular to the linear velocity $\vec{v}_r$ and $\vec{B}$. Therefore, a non-zero magnetic force may change the direction of the velocity and the dot product
$
\vec{F}_{ m } \cdot \vec{v}=q(\vec{v} \times \vec{B}) \cdot \vec{v}=0.
$
$\vec{F}_{\text {a }}$ perpendicular to the plane containing $\vec{v}$ and $\vec{B}$
But $\vec{F}_{ m } \cdot \vec{v}$ 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.
$
\vec{F}_{ m } \cdot \vec{v}=q(\vec{v} \times \vec{B}) \cdot \vec{v}=0.
$
$\vec{F}_{\text {a }}$ perpendicular to the plane containing $\vec{v}$ and $\vec{B}$
But $\vec{F}_{ m } \cdot \vec{v}$ 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.
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