Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

The magnetic force on a moving charge is one of the most fundamental known. Magnetic force is as important as the electrostatic or Coulomb force. Yet the magnetic force is more complex, in both the number of factors that affects it and in its direction, than the relatively simple Coulomb force.

The magnitude of the magnetic force $F$ on a charge depends on: the quantity of charge $q$, its speed $v$, the strength of magnetic field $B$, and the direction of motion relative to the magnetic field’s direction. Motion, and its direction, are critical.

In that situation, the magnitude of the magnetic force is The minimum force occurs when the direction of motion and the magnetic field’s direction are parallel to one another (i.e. zero or 180 degree angle between directions). In that situation, the magnitude of the magnetic force is

We define the magnetic field strength $B$ in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength $B$ is called the tesla (T) after the eccentric but brilliant inventor Nikola Tesla (1856–1943). To determine how the tesla relates to other SI units, we solve for the magnetic field strength.

So, the tesla is

(note that C/s = A).

Another smaller unit, called the gauss (G), where $\mathrm{1\; G}={\text{10}}^{-4}\mathrm{T}$, is sometimes used. The strongest permanent magnets have fields near 2 T; superconducting electromagnets may attain 10 T or more. The Earth’s magnetic field on its surface is only about $5\times {\text{10}}^{-5}\mathrm{T}$, or 0.5 G.

The direction of the magnetic force $\mathrm{\text{F}}$ is perpendicular to the plane formed by $\mathrm{\text{v}}$ and $\mathrm{\text{B}}$, as determined by the right hand rule 1 (or RHR-1), which is illustrated in Figure 10.5.1. RHR-1 states that, to determine the direction of the magnetic force on a positive moving charge, you point the thumb of the right hand in the direction of $\mathrm{\text{v}}$, the fingers in the direction of $\mathrm{\text{B}}$, and a perpendicular to the palm points in the direction of $\mathrm{\text{F}}$. One way to remember this is that there is one velocity, and so the thumb represents it. There are many field lines, and so the fingers represent them. The force is in the direction you would push with your palm. The force on a negative charge is in exactly the opposite direction to that on a positive charge.