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From Magnetic Field to Magnetic Force

As mentioned previously, only moving charged particles can interact with a magnetic field. Stationary electric charges are completely oblivious to the presence of magnetic fields. The magnetic force (also known as Lorentz force) on a moving electric charge is given by the relation,

$\vec{F}=q \left (\vec{v} \times \vec{B} \right )$

where

• $$q$$ is the charge on the particle of interest,
• $$\vec{v}$$ the velocity of the particle of interest,
• and $$\vec{B}$$ is the net magnetic field at the location of the particle of interest (created by all of the other moving charge particles in the universe).

Thus, the direction of the magnetic force on a moving charge is more complicated than in the analogous case of the electric force. In addition to determining the magnetic field at the location of the particle, you must know its velocity and perform a vector cross-product.

Magnetic Force on a Current-Carrying Wire

In many cases, instead of considering moving electric charges individually we will focus our attention on the collection of moving charges that make up an electric current. In an electric current, consider a small amount of charge, $$\mathrm{d} q$$, moving with velocity $$\vec{v}$$. This velocity can be represented as:

$\vec{v}=\frac{\mathrm{d} \vec{l}}{\mathrm{d} t}$

where $$\vec{l}$$ is the instantaneous displacement of this small collection of charge. Based on this observation, we can calculate the force acting on this small amount of charge by:

$\mathrm{d} \vec{F} = \left ( \mathrm{d} q \right ) \vec{v} \times \vec{B}$

$\mathrm{d} \vec{F} = \left ( \mathrm{d} q \right ) \frac{\mathrm{d} \vec{l}}{\mathrm{d} t} \times \vec{B}$

$\mathrm{d} \vec{F} = \left ( \frac {\mathrm{d} q}{\mathrm{d} t} \right ) \mathrm{d} \vec{l} \times \vec{B}$

Since the current in a wire is the amount of charge passing through any cross-section of the wire per second,

$\mathrm{d} \vec{F} = i \left ( \mathrm{d} \vec{l} \right ) \times \vec{B}$

Therefore the magnetic force on the entire current-carrying wire is given by the relation,

$\vec{F} = \int i \left ( \mathrm{d} \vec{l} \right ) \times \vec{B}$

where

• $$i$$ is the current in the wire of interest,
• $$\mathrm{d} \vec{l}$$ is an infinitesimal length of the wire of interest,
• $$\mathrm{d} \vec{B}$$ is the net magnetic field at the location of $$\mathrm{d} \vec{l}$$ (created by all of the other moving charged particles and currents in the universe),
• and the integral is over the entire length of the wire.