Controlled Thermonuclear Fusion — Question 5

Question 5

A magnetic field bends ion motion without supplying kinetic energy

A singly charged deuteron has mass $3.34\times10^{-27}$ kg and perpendicular speed $10^6$ m/s in uniform B=5.00 T. Use charge $1.60\times10^{-19}$ C and neglect collisions.

1. Find gyro-radius and gyro-period.

2. Show that the magnetic field does no work, and explain why a small orbit radius alone does not prove long-term plasma confinement.

Question 5 - Solution

A magnetic field bends ion motion without supplying kinetic energy

(a) Method and calculation

$$|q|v_\perp B=mv_\perp^2/r_L\Rightarrow\boxed{r_L=\frac{mv_\perp}{|q|B}=4.175\times10^{-3}\,\mathrm m}.$$$$\boxed{T_g=\frac{2\pi m}{|q|B}=2.62323\times10^{-8}\,\mathrm s}.$$

(b) Interpretation and further calculation

$$\boxed{\frac{dK}{dt}=q(\mathbf v\times\mathbf B)\cdot\mathbf v=0}.$$$$\mathbf F\cdot\hat{\mathbf B}=0\Rightarrow v_\parallel\text{ is not stopped by a uniform magnetic field}.$$

Physical check

Particles can stream along open field lines. Collisions, drifts and collective instabilities also affect confinement beyond the single-particle orbit.