Magnetic Force

The magnetic force is a resultant Force that occurs when two Charges are moving (non-static).

Definition §

Assume we have two moving Point Charges, $q_1$ and $q_2$, moving parallelly with velocities ${\vec{v}}_1$ and ${\vec{v}}_2$. Then the magnetic field is given by:

Formula - 1 §

Magnetic Force from two moving point charges via Biot-Savart Law

$${{\vec{F}}_B}_{q_2\to q_1}=\frac{{\mu }_0}{4\pi }\frac{q_1{q}_2}{r^2}{\vec{v}}_1\times (\overrightarrow{v_2}\times \hat{u})$$

Formula - 2 §

Magnetic Force from Magnetic Field

\vec{F}_{B} = q \vec{v} \times \vec{B} = Bvq \sin (\theta)

Terms §

• ${\vec{F}}_B$ = Magnetic force (in $\text{N}$)
  • ${{\vec{F}}_B}_{q_2\to q_1}$: Magnetic force of charge $q_2$ onto $q_1$
• ${\mu }_0$ = Permeability of free space $\approx 1.26\times 10^{-6}$ (in $N/A^2$)
• $q_1,q_2$ = Charges
• $\vec{B}$ = Magnetic Field
• $r$ = Distance between charges
• ${\vec{v}}_1,{\vec{v}}_2$ = Velocities on $q_1,q_2$ respectively
• $\hat{u}$ = Unit vector from $q_2$ to $q_1$

Implications §

The vector triple product, (two Cross Products together) has some interesting implications when it comes to the magnetic field. Cross products are zero when vectors are parallel, so any time a point charge is moving towards another point charge, it exerts no magnetic force on it. This also shows to violates newton’s third law, as force pairs that are not opposite in direction exist in this configuration:

Lastly, it’s useful to see how two different charges, moving together can superpose a magnetic force, through symmetry.

Relation to Magnetic Field §

Both magnetic field and force are interconnected, just like Electric Field and Electric Force are connected:

Specifically, this is the force exerted on a test charge of charge $q$ moving in a magnetic field $\vec{B}$ with velocity $\vec{v}$.