Electric and magnetic forces both affect the trajectory of charged particles, but in qualitatively different ways.

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.

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

$\text{F}=\text{qE}$

Where F is the force vector, q is the charge, and E is the electric field vector. Note that the direction of F is identical to E in 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 distance r so that:

It should be emphasized that the electric force F acts parallel to the electric field E. The curl of the electric force is zero, i.e.:

$▽\times \text{E}=0$

A consequence of this is that the electric field may do work and a charge in a pure electric field will 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:

$\text{F}=\text{qv}\times \text{B}=\text{qvBsin}\theta$

where B is the magnetic field vector, v is 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.

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 fields do not net work, since the particle motion is circular and therefore ends up in the same place. We express this mathematically as:

$\text{W}=\oint \text{B}\cdot \text{dr}=0$

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 charge q moves with velocity v in the presence of an electric field E and a magnetic field B, then it will experience a force:

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.

If multiple charges are involved, field lines are generated on positive charges, and terminate on negative ones.

In the case of magnets, field lines are generated on the north pole (+) and terminate on the south pole (-) – see the below figure. Magnetic ‘charges’, however, always come in pairs – there are no magnetic monopoles (isolated north or south poles). The curl of a magnetic field generated by a conventional magnet is therefore always non zero. Charged particles will spiral around these field lines, as long as the particles have some non-zero component of velocity directed perpendicular to the field lines.

A magnetic field may also be generated by a current with the field lines envisioned as concentric circles around the current-carrying wire. The magnetic force at any point in this case can be determined with the right hand rule, and will be perpendicular to both the current and the magnetic field.

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Key Points

• Magnetic Force: F = qvBsinθ

• Lorentz Force: F = q(E + vB)

• Right-hand Rule. The thumb points in the direction of the motion of charge, while the fingers follow the magnetic field lines. The palm faces the direction of the magnetic force.

• The force on a charged particle due to an electric field is directed parallel to the electric field vector in the case of a positive charge, and anti-parallel in the case of a negative charge. It does not depend on the velocity of the particle.

• In contrast, the magnetic force on a charge particle is orthogonal to the magnetic field vector, and depends on the velocity of the particle. The right hand rule can be used to determine the direction of the force.

• An electric field may do work on a charged particle, while a magnetic field does no work.

• The Lorentz force is the combination of the electric and magnetic force, which are often considered together for practical applications.

• Electric field lines are generated on positive charges and terminate on negative ones. The field lines of an isolated charge are directly radially outward. The electric field is tangent to these lines.

• Magnetic field lines, in the case of a magnet, are generated at the north pole and terminate on a south pole. Magnetic poles do not exist in isolation. Like in the case of electric field lines, the magnetic field is tangent to the field lines. Charged particles will spiral around these field lines.

Key Terms

Magnetic field lines: A graphical representation of the magnitude and the direction of a magnetic field.

Velocity: the rate of change of displacement with respect to time, velocity is a vector with magnitude and direction

Point charge: an electric charge regarded as concentrated in a mathematical point, without spatial extent.

Orthogonal: Of two objects, at right angles; perpendicular to each other.

Right hand rule: to determine the direction of the magnetic force on a positive moving charge, ƒ, point the thumb of the right hand in the direction of v, the fingers in the direction of B, and a perpendicular to the palm points in the direction of F.