Force on moving charge on a magnetic field (Lorentz Force)

Magnetic force can cause a charged particle to move in a circular or spiral path. Cosmic rays are energetic charged particles in outer space, some of which approach the Earth. They can be forced into spiral paths by the Earth’s magnetic field. Protons in giant accelerators are kept in a circular path by magnetic force. The bubble chamber photograph in [link] shows charged particles moving in such curved paths. The curved paths of charged particles in magnetic fields are the basis of a number of phenomena and can even be used analytically, such as in a mass spectrometer.

Consider a charged particle=q is moving with velocity v on a magnetic field B can experience magnetic force FM. If the direction of the magnetic field is represented by Y-axis, the velocity of the charged particle is along the XY plane making an angle θ with the direction of the magnetic field (B), then the direction of the magnetic force is represented by Z-axis.

Force-on-moving-charge-on-a-magnetic-field experimentally it is found that the magnitude of magnetic force is directly proportional to the magnitude of the
charge i.e. directly proportional to the velocity of the charge i.e directly proportional to the strength of the magnetic field

I .e directly proportional to the sine of the angle between v and B i.e

Now from the above equations, we get

This is the expression for Lorentz magnetic force where FM is perpendicular to the plane of v and B