Lorentz force and fine-beam tube

What is it about?

A charge moving perpendicular to a magnetic field feels the Lorentz force $F = q \cdot v \cdot B$, perpendicular to $v$ and $B$. For electrons the left-hand rule gives the direction (thumb: motion, index finger: field, middle finger: force). The force only changes the direction, so the electrons move on a circle.

• Acceleration: $v = \sqrt{2 \cdot (e/m) \cdot U}$.
• Helmholtz coils: $B = (4/5)^{3/2} \cdot \mu_0 \cdot n \cdot I / R$ with $n = 130$, $R = 15\,\text{cm}$.
• Circular path: $e v B = m v^2 / r$, so $r = \frac{1}{B}\sqrt{2 U m / e}$ and $e/m = \frac{2U}{B^2 r^2}$.

The marks on the ladder give the diameter (2 to 10 cm). Set the beam to a mark, record data points and plot $2U$ against $B^2 r^2$: the slope is $e/m \approx 1.76 \cdot 10^{11}\,\text{C/kg}$.

Try it

1. Without magnetic field. No field
2. Higher voltage, same current. Example
3. Left-hand rule. Left hand
4. Strong field. Strong field

Tasks

$n = 130$, $R = 15\,\text{cm}$, $I = 1.0\,\text{A}$. Find $B$.

$U = 200\,\text{V}$, $B = 1.0\,\text{mT}$. Find $v$ and $r$.

At $U = 250\,\text{V}$ the beam hits the 8 cm mark with $I = 1.71\,\text{A}$. Find $e/m$.

Ideas for the classroom

• Common stumbling blocks: “The Lorentz force speeds the electrons up.” – It only changes the direction. Diameter vs. radius: the marks give the diameter.