2024-04-15 Liénard-Wiechert potentials

Results from last classes

• Moving electrons radiate electromagnetic fields
  • Oscillating Electric Dipole example
• Radiation problem’s take the form of an inhomogenous wave equation
• Relation between electric and magnetic fields and potentials

$$\vec{E}=-\vec{\nabla }V-\frac{\partial \vec{A}}{\partial t}$$ $$\vec{B}=\vec{\nabla }\times \vec{A}$$

• Solutions to inhomogenous wave equation are retarded potentials
• Using approximations from retarded potential to find electric and magnetic potential
• Using potentials we can calculate fields. Radiation fields typically take the form of spherical waves. At large distances appears to be plane waves.
• From this, we calculate the poynting vector and then we can integrate to find power.

Liénard-Wiechert potentials

• Used to calculate fields from accelerating electrons

Usage

• Accelerated electrons are often used to produce electromagnetic radiation
• A commonly used example is microwave ovens
• Often used for research in accelerators, with both electrons and protons.
• Accelerating electrically charged particles circularly can create a source of EM radiation

Deriving Electric field

$$\vec{E}(\vec{r},t)=\frac{q}{4\pi {ϵ}_0}\frac{r}{(r\cdot \overrightarrow{\stackrel{⃗}{u}}{)}^3}[(c^2-v^2)\vec{u}+r\times (\vec{u}\times \vec{a})]$$

where $\vec{a}$ is the acceleration, $\vec{u}$ is the velocity of the charge and $\vec{r}$ is the vector to the field point

The electric field lines become distorted as the electron speed approaches light speed ($c$). The Coulomb field begins to concentrate perpendicular to the direction of motion.

11.12 9.109×10−31 kg is electron weight