Energy

Energy is measured in joules (J), it’s a conserved quantity.

$$w={\int }_{t_1}^{t_2}Pdt$$

i.e., the time-integral of electrical power.

Classical mechanics

The kinetic energy $T$ is the total work that must be done to bring it from a state of rest to some velocity.

$$T=\frac{1}{2}m{v}^2$$

The potential energy $V$ is work that may be converted into energy. For gravitational potential energy:

$$V=mgh$$ $$\Delta V=mg\Delta h$$

For a spring, we have recoverable elastic potential energy:

$$V_e=\frac{1}{2}k{x}^2$$ $$\Delta V_e=\frac{1}{2}k(x_2^2-x_1^2)$$

Electromagnetic energy

The electronvolt is the work done when a charge equal to one electron is taken across a potential difference of one volt. The idea is that the joule is often too large of a unit for measuring microscopic phenomena.

$$1eV=1.6\times 10^{-19}J$$

When we think about electromagnetic energy, we think about the energy stored in dynamic elements like capacitors or inductors.

Electric field

In the electric field case, we think about the energy stored in a capacitor. We first think about the amount of work to transfer an infinitesimal charge from one conductor to another:

$$d{W}_e=vdq=\frac{q}{C}dq$$

Then, we get a full expression for the electrostatic potential energy:

$$W_e=\frac{1}{2}{\iiint }_v{\rho }_{v}Vdv=\frac{1}{2}{\iiint }_v\vec{D}\cdot \vec{E}dv={\iiint }_v{w}_{e}dv$$

We define the electric energy density as the electric energy per unit volume:

$$w_e=\frac{1}{2}ϵE^2$$

Magnetic field

We follow a similar process for the magnetic field case with an inductor:

$$d{W}_m=vdq=L\frac{di}{dt}dq=Lidi$$

Our full expression is:

$$W_m={\int }_0^{I_0}Lidi={\iiint }_v\frac{1}{2}\vec{B}\cdot \vec{H}dv={\iiint }_v\frac{1}{2}\mu H^{2}dv$$

We will define the integrand as the magnetic energy density, the magnetic energy per unit volume.

$$w_m=\frac{1}{2}\mu H^2$$