• In electrostatics E-fields are conservative, meaning the work done in a moving charge only depends on the start and end points
• Work for a particle can be expressed as:

$$ W=q\int \vec{E}\cdot\text d\vec{l} $$

• For any closed loop the total work is zero:

$$ \oint_C \vec E \cdot \text d \vec l =0 $$

• $\oint_C$ is the line integral round the closed loop $C$

• When a charge moves along an electric field line from A to B the E-field will do work
• When then moving the charge from B to A we are converting the energy of the E-field to potential energy
• The potential energy per charge is the electric potential $\phi$ (voltage)

The scalar potential field

• We can create a scalar field that represents the electric potential $\phi$ from a point A to any point in space in the E-field. Generally A is set to be infinity and we get the following equation

$$ \phi(r)=-\int^r_\infin \vec E \cdot \text d \vec l $$

• To measure the potential from 2 points A and B we do:

$$ \Delta\phi=\phi(\vec r_B)-\phi(\vec r_A)=-\int^{r_A}_{r_B} \vec E \cdot \text d \vec l $$

<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/puzzle-piece_1f9e9.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/puzzle-piece_1f9e9.png" width="40px" /> Equipotential surfaces: Points in space that have the same $\phi$ are called equipotential surfaces

• No work is done moving a charge along an equipotential

• E-fields are normal to equipotential

• The closer equipotential are to each other the stronger the E-field </aside>

• The potential formula can be re-written to get the E-field:

$$ \vec E=-\vec \Delta \phi=-\left( \frac{\partial\phi}{\partial x}\hat{i}+\frac{\partial\phi}{\partial y}\hat{j}+\frac{\partial\phi}{\partial z}\hat{k} \right) $$