Categories: Electromagnetism, Physics, Plasma physics.

Lorentz force

The Lorentz force is an empirical force used to define the electric field \(\vb{E}\) and magnetic field \(\vb{B}\). For a particle with charge \(q\) moving with velocity \(\vb{u}\), the Lorentz force \(\vb{F}\) is given by:

\[\begin{aligned} \boxed{ \vb{F} = q (\vb{E} + \vb{u} \cross \vb{B}) } \end{aligned}\]

Uniform electric field

Consider the simple case of an electric field \(\vb{E}\) that is uniform in all of space. In the absence of a magnetic field \(\vb{B} = 0\) and any other forces, Newton’s second law states:

\[\begin{aligned} \vb{F} = m \dv{\vb{u}}{t} = q \vb{E} \end{aligned}\]

This is straightforward to integrate in time, for a given initial velocity vector \(\vb{u}_0\):

\[\begin{aligned} \vb{u}(t) = \frac{q}{m} \vb{E} t + \vb{u}_0 \end{aligned}\]

And then the particle’s position \(\vb{x}(t)\) is found be integrating once more, with \(\vb{x}(0) = \vb{x}_0\):

\[\begin{aligned} \boxed{ \vb{x}(t) = \frac{q}{2 m} \vb{E} t^2 + \vb{u}_0 t + \vb{x}_0 } \end{aligned}\]

In summary, unsurprisingly, a uniform electric field \(\vb{E}\) accelerates the particle with a constant force \(\vb{F} = q \vb{E}\). Note that the direction depends on the sign of \(q\).

Uniform magnetic field

Consider the simple case of a uniform magnetic field \(\vb{B} = (0, 0, B)\) in the \(z\)-direction, without an electric field \(\vb{E} = 0\). If there are no other forces, Newton’s second law states:

\[\begin{aligned} \vb{F} = m \dv{\vb{u}}{t} = q \vb{u} \cross \vb{B} \end{aligned}\]

Evaluating the cross product yields three coupled equations for the components of \(\vb{u}\):

\[\begin{aligned} \dv{u_x}{t} = \frac{q B}{m} u_y \qquad \quad \dv{u_y}{t} = - \frac{q B}{m} u_x \qquad \quad \dv{u_z}{t} = 0 \end{aligned}\]

Differentiating the first equation with respect to \(t\), and substituting \(\dv*{u_y}{t}\) from the second, we arrive at the following harmonic oscillator:

\[\begin{aligned} \dv[2]{u_x}{t} = - \omega_c^2 u_x \end{aligned}\]

Where we have defined the cyclotron frequency \(\omega_c\) as follows, which may be negative:

\[\begin{aligned} \boxed{ \omega_c \equiv \frac{q B}{m} } \end{aligned}\]

Suppose we choose our initial conditions so that the solution for \(u_x(t)\) is given by:

\[\begin{aligned} u_x(t) = u_\perp \cos\!(\omega_c t) \end{aligned}\]

Where \(u_\perp \equiv \sqrt{u_x^2 + u_y^2}\) is the constant total transverse velocity. Then \(u_y(t)\) is found to be:

\[\begin{aligned} u_y(t) = \frac{m}{q B} \dv{u_x}{t} = - \frac{m \omega_c}{q B} u_\perp \sin\!(\omega_c t) = - u_\perp \sin\!(\omega_c t) \end{aligned}\]

This means that the particle moves in a circle, in a direction determined by the sign of \(\omega_c\).

Integrating the velocity yields the position, where we refer to the integration constants \(x_{gc}\) and \(y_{gc}\) as the guiding center, around which the particle orbits or gyrates:

\[\begin{aligned} x(t) = \frac{u_\perp}{\omega_c} \sin\!(\omega_c t) + x_{gc} \qquad \quad y(t) = \frac{u_\perp}{\omega_c} \cos\!(\omega_c t) + y_{gc} \end{aligned}\]

The radius of this orbit is known as the Larmor radius or gyroradius \(r_L\), given by:

\[\begin{aligned} \boxed{ r_L \equiv \frac{u_\perp}{|\omega_c|} = \frac{m u_\perp}{|q| B} } \end{aligned}\]

Finally, it is easy to integrate the equation for the \(z\)-axis velocity \(u_z\), which is conserved:

\[\begin{aligned} z(t) = z_{gc} = u_z t + z_0 \end{aligned}\]

In conclusion, the particle’s motion parallel to \(\vb{B}\) is not affected by the magnetic field, while its motion perpendicular to \(\vb{B}\) is circular around an imaginary guiding center. The end result is that particles follow a helical path when moving through a uniform magnetic field:

\[\begin{aligned} \boxed{ \vb{x}(t) = \frac{u_\perp}{\omega_c} \begin{pmatrix} \sin\!(\omega_c t) \\ \cos\!(\omega_c t) \\ 0 \end{pmatrix} + \vb{x}_{gc}(t) } \end{aligned}\]

Where \(\vb{x}_{gc}(t) \equiv (x_{gc}, y_{gc}, z_{gc})\) is the position of the guiding center. For a detailed look at how \(\vb{B}\) and \(\vb{E}\) can affect the guiding center’s motion, see guiding center theory.

References

  1. F.F. Chen, Introduction to plasma physics and controlled fusion, 3rd edition, Springer.

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