Charged Particles in Magnetic Fields

A charged particle moving in an existing Magnetic Field experiences a Magnetic Force which affects the motion, causing either circular or helical motion.

Velocity Perpendicular to Magnetic Field §

In the case where ${\vec{v}}_{particle}\perp \vec{B}$, we have a Magnetic Force perpendicular to the motion of the particle. This ultimately results in a circular path in regions of the magnetic field.

We know that magnetic force on a charged particle in a field can be given by:

We can see that in different ‘snapshots’, the particle always follows a circular path, as it always experiences an acceleration perpendicular to it’s velocity

This provides us with some useful formulae. Because we know that the magnetic force is always acting perpendicular to the velocity of the particle, it is a suitable candidate for Centripetal Force. Recall the definition a centripetal force:

We can equate the two forces:

$${\vec{F}}_B={\vec{F}}_c$$

$$qvB\sin (\theta )=\frac{m{v}^2}{r}$$

q\cancel{v}B\sin(\theta) = \dfrac{mv^\cancel{ 2 }}{r}

$$qB=\frac{mv}{r}$$

We can also find the frequency of the particle loop, which gives the number of loops done by the particle per second:

$$T=\frac{d}{s}=\frac{2\pi r}{v}$$

$$f=\frac{1}{T}$$

$$f=\frac{v}{2\pi r}=\frac{qB}{2\pi m}$$

Velocity not perpendicular to Magnetic Field §

In the case where ${\vec{v}}_{particle}\cancel{\perp }\vec{B}$ we have $v$ which can be split into two components using Vector Projection

$$\vec{v}={\vec{v}}_{\parallel }+{\vec{v}}_{\perp }$$

• ${\vec{v}}_{\parallel }$ causes a uniform straight motion of the particle, since it is parallel to the B-field, and as such $\vec{B}\times \vec{v}=0$
• ${\vec{v}}_{\perp }$ causes circular motion, like above.

This combination of translational and circular motion gives rise to a helical pattern.