# The Lorentz Force law

## Problem

The Lorentz force law tells us how a non-relativistic charged particle $P$ moves under the influence of electric and magnetic forces:

$$m{\bf a(t)} = q\left({\bf E(t)}+{\bf v(t)} \times {\bf B(t)}\right)$$

In this vector differential equation, ${\bf a}$ and ${\bf v}$ are the acceleration and velocity of the particle of mass $m$ and charge $q$; The vectors ${\bf E}$ and ${\bf B}$ are the electric and magnetic fields respectively.

The motion of the particle in various electric and magnetic fields is described. In each case, what can we deduce about the vectors ${\bf E}$ and ${\bf B}$? Can you find particular solutions for fields giving rise to each motion? Describe clearly the configuration of the fields, along with an initial velocity, paying attention to whether various quantities are parallel or orthogonal to each
other.

1) The particle sits at the origin and does not move.

2) The particle moves with a constant velocity, passing through the origin at time $t=0$.

3) The particle moves in a circle in the $x$-$y$ plane, centred on the origin.

4) The particle moves in a spiral up around the $z$-axis.

Once you have your solutions, could you see how to relate these back directly to the original differential equation?

Can you solve the equations for any other forms of ${\bf E}$ and ${\bf B}$?

Can you construct fields for which the kinetic energy of the particle will continually increase?

## Getting Started

In Cartesian coordinates, the position ${\bf r}$ of the charged particle at time $t$ is determined by three functions$x(t), y(t), z(t)$, so that $${\bf r} = (x(t), y(t), z(t))$$

In component form, the Lorentz force law becomes

$$\left(\frac{m}{q} \ddot{x}, \frac{m}{q} \ddot{y}, \frac{m}{q} \ddot{z}\right)

=

\left( E_x+\dot{y}B_z-\dot{z}B_y, E_y+\dot{z}B_x-\dot{x}B_z, E_z+\dot{x}B_y-\dot{y}B_x \right)

$$

You can then equate components to make three separate (simultaneous) differential equations.

Try differentiating these to solve for particular cases of ${\bf E}$ and ${\bf B}$.

You can make life easier for yourself by supposing the the electric and magnetic fields do not change over time.