0.6 Lorentz force

Lorentz force is the electromagnetic force on a point or test charge. The corresponding force law for electromagnetic force is an empirical law providing the combined expression for electrical and magnetic forces experienced by the test charge. Lorentz force for a point charge comes into existence under certain conditions. The existence of either electrical or magnetic or both fields is primary requirement.

The force law sets up the framework under which two force types (electrical and magnetic) operate. The law is fundamental to the study of electromagnetic interactions in terms of field concepts. For the consideration of force(s) on the test charge, the important deduction is that electrical field interacts only with electrical field and magnetic field interacts only with magnetic field. In our context of electromagnetic force, we can say that electrical force results from interaction of two electrical fields and magnetic force results from interaction of two magnetic fields.

Lorentz force expression

The law is stated in vector form as :

$$\mathbf{F}=q[\mathbf{E}+\left(\mathbf{v}X\mathbf{B}\right)]$$

We may recognize that Lorentz force is actually vector sum of two forces :

$$\Rightarrow \mathbf{F}=q\mathbf{E}+q\left(\mathbf{v}X\mathbf{B}\right)$$

For convenience, we refer the first force as Lorentz electrical force and second force as Lorentz magnetic force. The Lorentz electrical force is given by first part as :

$${\mathbf{F}}_E=q\mathbf{E}$$

The electrical part of law is actually the relation we have already studied in the context of Coulomb’s law and Electrical field. Electrical force on the point charge "q" acts in the direction of electrical field ( E ) and as such the particle carrying the charge is accelerated in the direction of E . If "m" be the mass of the particle carrying charge, then acceleration of the particle is :

$${\mathbf{a}}_E=\frac{{\mathbf{F}}_E}{m}=\frac{q\mathbf{E}}{m}$$

Lorentz magnetic force is given by second part as :

$${\mathbf{F}}_M=q\left(\mathbf{v}X\mathbf{B}\right)$$

Magnetic force on the point charge "q" acts in the direction perpendicular to the plane formed by v and B vectors. The direction of vector cross product is the direction of magnetic field, provided test charge is positive. The orientation of vector cross product is determined using Right hand thumb rule. If the curl of right hand follows the direction from vector v to B , then extended thumb points in the direction of vector cross product.

Direction of vector cross product

We should understand an important point that direction of magnetic field is determined not by the direction of vector cross product vXB alone, but by the direction of expression "q( vXB )". What it means that if charge is negative, then direction of force is opposite to that determined by vector cross product " vXB ". The figure below shows the opposite orientations of vector cross product " vXB " and the magnetic force.

Lorentz magnetic force

The acceleration of the particle is given by :

$${\mathbf{a}}_M=\frac{{\mathbf{F}}_M}{m}=\frac{q\left(\mathbf{v}X\mathbf{B}\right)}{m}$$

The magnitude of magnetic force is given by :

$$F_M=qvB\sin \theta$$

where θ is the smaller angle between v and B vectors. The magnitude of magnetic field is maximum when θ = 90 or 270 and the maximum value of magnetic field is qvB. It is also clear from the expression of magnitude that magnetic force is zero even when magnetic field exists for following cases :