0.6 Lorentz force  (Page 3/4)

Yet another important consequence of the nature of magnetic force is that a charged particle in magnetic field keeps changing direction of motion of the particle all the time. Since direction of velocity is changed every instant, direction of magnetic force being perpendicular to it is also changed all the time. Note that direction of magnetic force is automatically adjusted or changed with the motion. If the particle does not escape out of the magnetic field, the implication is that the particle may approximate a circular path. At any moment – whether particle completes a circular path or not – the magnetic force acts in radial direction to the motion. On a comparison note, we can see that the electric force is independent of the direction of motion. It is along electric field. It does not change with motion.

Lorentz magnetic force

We make use of this feature in many important applications like cyclotron to accelerate particle or entrapping plasma etc. But we should be aware of its role in these applications. The effect of Magnetic force is limited to change in direction only. Change in speed is effected by electric field.

Magnitude of magnetic force

The magnetic field is a weak field and so is the magnetic force. Let us consider an electron moving with a velocity $3X{10}^7$ m/s in a magnetic field of $5X{10}^{-3}$ T. If velocity and magnetic field are perpendicular to each other, then magnetic force on the electron is :

$$F_M=qvB=1.6X{10}^{-19}X3X{10}^{7}X5X{10}^{-3}=2.4X{10}^{-14}N$$

Clearly, magnetic force is really very weak. However, even this weak force is great enough for subatomic particle like electron. For example, the acceleration of electron due to this magnetic force is :

$$a=\frac{F_M}{m}=\frac{2.4X{10}^{-14}}{9.1X{10}^{-31}}=2.6X{10}^{16}m/s^2$$

Indeed this is an extraordinary acceleration.

Context of lorentz force law

Lorentz magnetic force law completes the picture on “effect side” in the study of electromagnetism. The “cause side” i.e. generation of magnetic field is described by Biot-Savart law. Thus, Lorentz force law describes the effect of electric and magnetic fields on a test charge – but not the cause of these fields. This is a serious limitation because test charge on its own is also the cause of electric and magnetic fields. These fields, in turn, would modify the fields operating on the test charge.

Also, the electromagnetic force causes acceleration of test charge. An accelerated charge, in turn, radiates. As such, application of Lorentz force law by itself would not be sufficient to describe motion of test charge. A charged electron which is expected to describe a circular motion under magnetic field without consideration of radiation would actually spiral down with radiation as shown in the figure and expected motion might simply be not there.

Motion of charge under magnetic field

Recall that this was the reason for which Rutherford’s atomic model was eventually rejected and Bohr’s model was accepted. We shall, however, ignore radiation while studying motion of charged particles under electromagnetic fields – unless state specifically to consider radiation.