Question

A uniform magnetic field exerts a force on a moving charge which is equal to the charge times the vector product of the velocity and the magnetic field. Can this force change the magnitude of the velocity of the charge? Explain. Can both uniform electric and magnetic fields exist (for non-zero fields) where the net force on a charge is zero? If so, what are the constraints on the fields? If not, why not?

Answer 1

Ans

a) The magnetic force on the moving charge is equal to the charge times the vector cross product of the velocity and the magnetic field. So the force is perpendicular to the velocity of the charge(and the magnetic field too). Since no component of the force is along the velocity of the charge, hence the magnitude of the velocity will not change.

Or since force is perpendicular to the displacement(velocity) of the charge, hence work done by the magnetic force on the charge is zero, So by work-kinetic energy theorem, the change in the kinetic energy of the charge = 0 J , hence speed of the charge cant be changed by the magnetic force on it.

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b) Yes both uniform electric and magnetic fields can exist (for non-zero fields) where the net force on a charge is zero.

Net force on the moving charge = 0 that means the electric force and magnetic force must be equal and in opposite direction. Since the electric force on a charge acts either along the electric field or opposite to the electic field. So Electric field and magetic field must be perpendicular to each other as magnetic force is perpendicular to the magnetic field.

The strength of the magnetic field => F_B = |q|vBsin()

is the angle between magnetic field (B) and velocity (v) of the charge.

The strength of the electric field => F_E = |q|E  

F_B = F_E as net force on the charge is zero

ie |q|vBsin() = |q|E => E = vBsin()

The relation between the fields => E = vBsin()

i)Special case when the ()angle between B and v = 90^o => E = vBsin(90^o) = vB

E = vB when = 90^o