# Motion in Combined Electric and Magnetic Field

When a moving charge is acted upon by electric and magnetic fields, the charge experiences a force called Lorentz force, which is a vector sum of forces due to electric and magnetic fields.

When the electric field, the magnetic field, and the motion of charge are mutually perpendicular to each other( as shown in the fig  )then they are called as crossed fields and forces due to electric and magnetic fields will act in the opposite directions so the Lorentz force F will be

F = qE \hat{i} + \left(q V \hat{i} \times B \hat{k} \right) = q E \hat{j} - q VB \hat{i} = q (E - VB) \hat{j}

When the strength of electric and magnetic fields are varied to get the forces due to electric and magnetic field to be equal then the charge can move in the field without any deflection

q E = qv B

V = E/B.

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Force on a Charged particle in Combined uniform Electric and Magnetic fields
\overrightarrow{F} = \overrightarrow{F}_{E} + \overrightarrow{F}_{M} = q\overrightarrow{E} + q(\overrightarrow{v} \times \overrightarrow{B}) = q (\overrightarrow{E} + \overrightarrow{v} \times \overrightarrow{B})