The magnetic force, always acting perpendicular to a charged particle's velocity, does no work on the particle, thus it cannot change the particle's speed or kinetic energy. This perpendicularity, determined by the cross product, only alters the particle's direction, causing it to follow curved or circular paths. Conversely, an electric field can exert a force parallel to the particle's motion, thereby performing work and changing the particle's speed and kinetic energy. For instance, particle accelerators utilize electric fields for increasing particle speed and magnetic fields for controlling the direction of the particle beam.
<aside> 🧄
$\complement\cdots$Counselor
</aside>
The most crucial concept is that the magnetic force ( $F_B$ ) is always perpendicular to the particle's velocity ( $v$ ). This is a defining characteristic of the magnetic force, which is determined by the cross-product ( $v \times B$ ).
Since work is defined as the component of force in the direction of motion, and the magnetic force has zero component in the direction of motion, no work can be done. Mathematically, this is shown by the dot product: $d W=F_B \cdot d x=0$. Because the force vector is perpendicular to the displacement vector at every instant, their dot product is always zero.
Because no work is done, a magnetic field cannot change the kinetic energy or speed of a charged particle. It can only change the particle's direction, causing it to move in a curved or circular path. It is the electric field that can accelerate the particle and thus change its speed and kinetic energy.
‣
<aside> 🧄
</aside>