Force due to both electric and magnetic forces will influence the motion of charged particles. However, the resulting change to the trajectory of the particles will differ qualitatively between the two forces. Below we will quickly review the two types of force and compare and contrast their effects on a charged particle.

Electrostatic Force and Magnetic Force on a Charged Particle

Recall that in a static, unchanging electric fieldEthe force on a particle with charge q will be:

F=qEF=qE

WhereFis the force vector, q is the charge, andEis the electric field vector. Note that the direction ofFis identical toEin the case of a positivist charge q, and in the opposite direction in the case of a negatively charged particle. This electric field may be established by a larger charge, Q, acting on the smaller charge q over a distancerso that:

E=∣∣Fq∣∣=k∣∣qQqr2∣∣=k|Q|r2E=|Fq|=k|qQqr2|=k|Q|r2

It should be emphasized that the electric forceFacts parallel to the electric fieldE. The curl of the electric force is zero, i.e.:

▽×E=0▽×E=0

A consequence of this is that the electric field may do work and a charge in a pure electric fieldwill follow the tangent of an electric field line.

In contrast, recall that the magnetic force on a charged particle is orthogonal to the magnetic field such that:

F=qv×B=qvBsinθF=qv×B=qvBsinθ

whereBis the magnetic field vector,vis the velocity of the particle and θ is the angle between the magnetic field and the particle velocity. The direction of F can be easily determined by the use of the right hand rule.[image]Right Hand Rule: Magnetic fields exert forces on moving charges. This force is one of the most basic known. The direction of the magnetic force on a moving charge is perpendicular to the plane formed by v and B and follows right hand rule–1 (RHR-1) as shown. The magnitude of the force is proportional to q, v, B, and the sine of the angle between v and B.

If the particle velocity happens to be aligned parallel to the magnetic field, or is zero, the magnetic force will be zero. This differs from the case of an electric field, where the particle velocity has no bearing, on any given instant, on the magnitude or direction of the electric force.

The angle dependence of the magnetic field also causes charged particles to move perpendicular to the magnetic field lines in a circular or helical fashion, while a particle in an electric field will move in a straight line along an electric field line.

A further difference between magnetic and electric forces is that magnetic fieldsdo not net work,since the particle motion is circular and therefore ends up in the same place. We express this mathematically as:

W=∮B⋅dr=0W=∮B⋅dr=0

Lorentz Force

The Lorentz force is the combined force on a charged particle due both electric and magnetic fields, which are often considered together for practical applications. If a particle of chargeqmoves with velocityvin the presence of an electric fieldEand a magnetic fieldB, then it will experience a force:

F=q[E+vBsinθ]F=q[E+vBsinθ]

Electric and Magnetic Field Lines

We mentioned briefly above that the motion of charged particles relative to the field lines differs depending on whether one is dealing with electric or magnetic fields. There are some notable differences between how electric and magnetic field lines are conceptualized. The electric field lines from a positive isolated charge are simply a sequence of evenly-spaced, radially directed lines pointed outwards from the charge. In the case of a negative charge, the direction of the field is reversed. The electric field is directed tangent to the field lines. Of course, we imagine the field lines are more densely packed the larger the charges are. One can see clearly that the curl of the electric force is zero.