Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S16 / 3

Take $\mathbf{B}=B\hat{\mathbf{z}}$. The magnetic Lorentz force givesThus $v_x+i{v}_y=(v_x(0)+i{v}_y(0))e^{-i\Omega t}$, while $v_z$ is constant. Integrating the transverse velocity gives a circle of radius $r_L=v_{\perp }/|\Omega |$ about a fixed guiding centre; adding $z(t)=z_0+v_{\parallel }t$ gives a helical motion in a uniform magnetic field. The magnetic force does no work, so the speed and kinetic energy remain constant. The handedness reverses with the sign of $q$. If $v_{\parallel }=0$ the helix degenerates to a circle; if $v_{\perp }=0$ it is a straight line parallel to the field.

Seek a constant transverse guiding-centre drift ${\mathbf{v}}_d$ whose magnetic Lorentz force balances the applied force. DefineSince $\mathbf{F}\cdot \mathbf{B}=0$, the vector triple-product identity gives $q{\mathbf{v}}_d\times \mathbf{B}=-\mathbf{F}$. Writing $\mathbf{v}={\mathbf{v}}_d+\mathbf{u}$ therefore reduces the full equation to $m\dot{\mathbf{u}}=q\mathbf{u}\times \mathbf{B}$, the ordinary gyromotion equation. Its transverse orbit is circular in the drifting frame, soAveraging over a gyroperiod removes the transverse oscillatory velocity and leaves this drift, together with any constant parallel velocity. For an electric force $\mathbf{F}=q\mathbf{E}$, this becomes the charge-independent $\mathbf{E}\times \mathbf{B}/B^2$ drift; a charge-independent force instead gives opposite drift directions for opposite charge signs.

A purely toroidal magnetic field is stronger on the inner side of the torus and its field lines are curved. The resulting magnetic gradient drift and curvature drift move ions and electrons vertically in opposite directions. Charge separation can then produce an electric field whose E-cross-B drift moves the plasma across the nominal toroidal confinement region. Closing field lines into circles therefore does not by itself guarantee confinement.

A tokamak combines a strong externally produced toroidal field with a poloidal field supplied largely by toroidal plasma current. Field lines wind helically around nested magnetic flux surfaces. A particle following a field line samples different vertical and radial parts of the torus, so the systematic displacements associated with the toroidal geometry can compensate over its orbit rather than accumulating as in the pure-toroidal case. The poloidal field also provides the magnetic structure needed for force balance. This improves confinement but does not eliminate all orbit losses or transport: trapped particles, collisions, turbulence and magnetohydrodynamic instabilities remain important. Maintaining the plasma current and suitable field-line winding is part of the confinement problem.

The supplied confinement criterion alone fixes a product, not a unique current or machine size. Choose a representative ion density $n_i=10^{20}{\mathrm{m}}^{-3}$. ThenFor $n_i=10^{19}$ to $10^{21}{\mathrm{m}}^{-3}$, the same criterion gives $\tau \gtrsim 10$ to $0.1\mathrm{s}$. These are illustrative density choices, not additional source data.

For a current estimate, assume a large-aspect-ratio circular tokamak with major radius $R_0=3\mathrm{m}$, minor radius $a=1\mathrm{m}$, toroidal field $B_{\varphi }=5\mathrm{T}$ and edge tokamak safety factor $q_s\simeq 3$. The geometrical winding relation and Ampère's law giveFor the stated assumptions,so toroidal plasma currents of order megaamperes are a natural scale. As a consistency check at $T\sim 10^8\mathrm{K}$, quasineutral singly charged heavy hydrogen has total thermal pressure $p\simeq 2n_i{k}_{B}T\simeq 2.8\times 10^5\mathrm{Pa}$. With $5\mathrm{T}$, $2{\mu }_{0}p/B^2\simeq 0.028$, so the assumed magnetic pressure exceeds the thermal pressure. The particular current depends on geometry and field-line pitch; it cannot be inferred from the confinement product alone.