A charge q moving with a velocity through a magnetic field of induction \(\vec{B}\) experiences a magnetic force perpendicular both to \(\vec{B}\) and .\(\vec{v}\)
Experimental observations show that the magnitude of the force is proportional to the magnitude of \(vec{B}\) the speed of the particle, the charge q and the sine of the angle θ between \(\vec{v}\) and \(\vec{B}\). That is, the magnetic force, Fm = qvB sin θ
∴ \(\vec{F_m}\) = q\((\vec{v}×\vec{B})\)
Therefore, at every instant \(vec{F_m}\) acts in a direction perpendicular to the plane of \(\vec{v}\) and \(\vec{B}\).
If the moving charge is negative, the direction of the force \(\vec{F_m}\) acting on it is opposite to that given by the right-handed screw rule for the cross-product \(\vec{v}×\vec{B}\) .
If the charged particle moves through a region of space where both electric and magnetic fields are present, both fields exert forces on the particle.
The force due to the electric field \(\vec{E}\) is \(\vec{F_e}\)= \(q\vec{E}\) .
The total force on a moving charge in electric and magnetic fields is called the Lorentz force :
\(\vec{F}\) =\(\vec{F_e}\)+\(\vec{F_m}\) = \(q(\vec{E}+\vec{v}×\vec{B})\) ……(1)
Special cases :
(i) \(\vec{v}\) is parallel or antiparallel to \(\vec{B}\) : In this case,
Fm = qvB sin 0° = 0. That is, the magnetic force on the charge is zero.
(ii) The charge is stationary (v = 0) : In this case, even if q ≠ 0 and B ≠ 0,
Fm = q(0)B sin θ = 0. That is, the magnetic force on a stationary charge is zero.
From Eq. (1) it may be observed that the force on the charge due to electric field depends on the strength of the electric field and the magnitude of the charge. However, the magnetic force depends on the velocity of the charge and the cross product of the velocity vector the magnetic field vector and the charge q
