Helmholtz Decomposition Theorem

Charge Density

Amount of electric charge per unit volume area or length, total charge in R = $\iiint_R \rho dV$

• Measured in Coloumbs per cubic meter $\frac{C}{m^3}$
• Represents how many charges are present per unit volume at one point
• You can calculate the total charge enclosed from the volume charge density

Spherical

Represents the amount of charge

Cylindrical

Planar

Dirac Distributions

Used to discuss the flow of current in very localized regions, such as the context of a point charge moving along a trajectory, or in the case of an infinitely thin wire where the current is confined to a line rather than spread out over a volume or area

1. Point Charge at $\vec r = (0,0,0)$

   Cartesian: $\rho (\vec r) = Q \delta(x) \delta (y) \delta (z) → \text{total} = \iiint Q \delta (x) \delta (y) \delta (z) dxdydz = Q$

   Spherical: $\rho (r)=\frac{Q}{2 \pi r^2} \delta (r) \rightarrow \text{total} = \iiint \frac{Q}{2 \pi r^2} \delta (r) r^2 sin \phi dr d\phi d \theta = 2Q \int \delta (r)dr = Q$

2. Line Charge along z-axis:

   Cartesian: $\rho(\vec r) = \lambda \delta (x) \delta (y)$

   Cylindrical: $\rho (r)=\frac{\lambda}{\pi r} \delta (r)$

3. Infinite Plane

   $\rho (z) = \sigma \delta (z)$

4. Spherical Shell

   $\rho (r)=\sigma \delta (r-R)$

5. Cylindrical Shell

   $\rho (r)=\sigma \delta (r-R)$

Current Flux Density $(\vec J)$

Vector quantity that describes the amount of electric current flowing through a unit area of cross-section