Magnetic Field

A bar magnet is one of the sources of magnetic field. The bar magnet consists of two poles, a north pole (N) and a south pole (S):

When holding two magnets close to each other, the like poles will repel each other while the opposite poles attract.

Magnetic poles always exist in pair. When a bar magnet is broken, two new bar magnets are obtained:

The electric field is the force per unit charge:

\boldsymbol{E} = \frac{\boldsymbol{F_e}}{q},

however, due to the absence of magnetic monopoles, the magnetic field $\boldsymbol{B}$ must be defined differently.

To define it, consider a particle of charge $q$ , moving at a velocity $\boldsymbol{v}$ . Experimentally, it was found:

The magnitude of the magnetic force $\boldsymbol{F_B}$ is proportional to both $v$ and $q$ .
The magnitude and direction of $\boldsymbol{F_B}$ depends on $\boldsymbol{v}$ and $\boldsymbol{B}$ .
$\boldsymbol{F_B}$ vanishes when $\boldsymbol{v}$ is parallel to $\boldsymbol{B}$ . When $\boldsymbol{v}$ and $\boldsymbol{B}$ forms an angle $\theta$ , $\boldsymbol{F_B}$ is perpendicular to a plane formed by them and the magnitude of $\boldsymbol{F_B}$ is proportional to $\sin \theta$ .
When the sign of $q$ switches, the direction of $\boldsymbol{F_B}$ reverses.

Plot of magnetic field and magnetic force

The observations can be summarized into:

\begin{align*} \boldsymbol{F_B} &= q\boldsymbol{v} \times \boldsymbol{B}, \\ F_B &= |q| v B \sin \theta. \end{align*}

The unit of magnetic field is tesla ( $T$ ). From $B = \frac{F_B}{|q|v}$ , tesla is equal to:

T = \frac{N}{C\ m\ s^{-1}} = \frac{N}{A\ m}.

Another commonly used unit is gauss ( $G$ ), where $1 T = 10^4 G$ .

Note that $\boldsymbol{F_B}$ is always perpendicular to the plane formed by $\boldsymbol{v}$ and $\boldsymbol{B}$ and cannot change the particle's speed $v$ . $\boldsymbol{F_B}$ does no work on the particle:

dW = \boldsymbol{F_B} \cdot d\boldsymbol{s} = q(\boldsymbol{v} \times \boldsymbol{v}) \cdot \boldsymbol{B}\ dt = 0.

However, the direction of $\boldsymbol{v}$ can be changed by $\boldsymbol{F_B}$ .

Magnetic Force on a Current

A charged particle moving at a speed $\boldsymbol{v}$ experiences a magnetic force $\boldsymbol{F_B}$ . Electric current is a collection of moving charged particles, meaning a wire carrying current also experiences the magnetic force.

Consider a long straight wire suspended in the region between two magnetic poles. The magnetic field is illustrated by the black dots and flows out of the page. When a downward current is present, the wire is deflected to the left. However, when an upward current is present, the wire is deflected to the right:

Consider a segment of wire of length $\ell$ and cross-sectional area $A$ . The magnetic field points into the page:

The charges move at an average drift velocity $\boldsymbol{v_d}$ and the total charge in this segment is:

Q = q n A \ell,

where $n$ is the number of charges per unit volume. The total magnetic force on the segment is:

\boldsymbol{F_B} = Q \boldsymbol{v_d} \times \boldsymbol{B} = q n A l (\boldsymbol{v_d} \times \boldsymbol{B}) = I (\boldsymbol{\ell} \times \boldsymbol{B}),

where $I = n q v_d A$ and $\boldsymbol{\ell}$ is vector with magnitude $\ell$ and with the same direction as the electric current.

For a wire of arbitrary shape, the magnetic force is obtained by summing over the force acting on the small segments $d\boldsymbol{s}$ :

\begin{align*} d \boldsymbol{F_B} &= I d\boldsymbol{s} \times \boldsymbol{B}, \\ \boldsymbol{F_B} &= I \boldsymbol{B} \times \int_C d\boldsymbol{s}. \end{align*}

For a closed loop, the sum of the segments is zero, and thus the total force exerted is zero:

\boldsymbol{F_B} = \boldsymbol{F_B} = I \boldsymbol{B} \times \oint_C d\boldsymbol{s} = 0.