Lorentz Force

The total force experienced by a charged particle moving with velocity \[\vec v\] in the presence of both an electric field \[\vec E\] and a magnetic field \[\vec B\] simultaneously.

\[\vec{F}=q\vec{E}+q(\vec{v}\times\vec{B})\]

Electric Force on a Charge

• Acts on the charge whether it is stationary or moving.

• Direction: along \[\vec E\] for positive charge, opposite for negative charge.

• Formula: \[\vec F_E\] = q\[\vec E\]

Magnetic Force on a Charge

• Acts only when the particle is moving.
• Direction: perpendicular to both \[\vec v\] and \[\vec B\] (right-hand rule).
• Formula: \[\vec F_B\] = q(\[\vec v\] × \[\vec B\])
• Does no work on the particle — it changes direction, not speed.

When \[\vec E\] and \[\vec B\] act perpendicular to each other and to \[\vec v\], the particle can travel undeflected if the electric and magnetic forces balance.

Condition for no deflection:

This principle is the basis of a velocity selector, used in mass spectrometers to filter particles of a specific velocity before measuring their specific charge q/m.

Problem: An electron is accelerated through a potential difference of 2.0 kV and enters a magnetic field of 0.15 T.