Time Varying E and B Field

Time-Varying E Field

The electric field

$$\mathbf E=\mathbf E_0 e^{i\omega t}\mathbf{\hat x}$$

The polarization drift

$$\mathbf v_p=\pm\frac{1}{\omega_c B}\frac{dE}{dt}$$

Time Varying B Field

The magnetic field

$$\nabla\times\mathbf E=-\mathbf{\dot B}$$

The magnetic moment is invariant in slowly varying magnetic fields

$$\delta\mu=0$$

The magnetic flux through a Larmor orbit is constant

$$\Phi=B\pi\frac{v_\perp^2}{\omega_c^2}=B\pi\frac{v_\perp^2m^2}{q^2B^2}=\frac{2\pi m}{q^2}\frac{\frac{1}{2}mv_\perp^2}{B}=\frac{2\pi m}{q^2}\mu$$

• $\Phi$ is constant if $\mu$ is constant.

The First Adiabatic Invariant, $\mu$

The Second Adiabatic Invariant, J

The Third Adiabatic Invariant, $\Phi$