Notebook

# Electromagnetic Fields

## Chapter 2: Conductivity, Resistivity and Capacitance

### Conductivity and Resistivity

When an electric field is applied to a material, charges experience force and create current. The ease at which these charges move is determined by the relationship between the current density and electric field, $J=\sigma.E$, with $\sigma$ meaning conductivity and measured in Siemens/metre.

Conductivity can also be represented as $\cfrac{1}{\rho}$ where $\rho$ is called resistivity. The internal properties of the material change with external conditions. The relationship with temperature can be approximated linearly as $\rho = \rho_0.(1+\alpha.(T_1-T_0))$, where alpha is the fixed temperature coefficient, $T_0$ is fixed room temperature, $\rho_0$ is the resistivity at temperature $T_0$, and $T_1$ is the current temperature.

Conductivity and resistivity are intrinsic properties of the material, they do not change with the shape or length of the material.

### Ohm's Law

For a straight object the following formula can be rearranged:

$J=\sigma.E$

$\cfrac{E}{J} = \rho = \cfrac{\cfrac{V}{l}}{\cfrac{I}{A}}$

$\cfrac{V}{I} = \rho.\cfrac{l}{A} = R$

The relationship V/I is called resistance and is known as Ohm's Law. It depends on the intrinsic resistivity of the material, its length and cross-section. Resistance is measured in Ohms. The formula above shows the resistance for straight objects and is generally used for wires.

Materials that obey this law are called ohmic.

### Capacitance

An isolated conductor has a property called self-capacitance, which is defined as the amount of electric charge needed to raise the conductor's electric potential by one Volt. The reference point for this potential is a theoretical conducting sphere with infinite radius, with the conductor centred inside this sphere. This is represented as $C={\cfrac {q}{V}}$ measured in Farads.

There is also mutual capacitance which is the capacitance between two conductors. The most common form is a parallel-plate capacitor. For a capacitors with plate charges +q and -q and potential difference V the same formula applies:

$C=\cfrac{q}{V}=\cfrac{\cfrac{dq}{dt}}{\cfrac{dV}{dt}} = \cfrac{I}{V'}$, transforming q to I

$C.V' = I$,

$C.V'.V = I.V = P$, converting to Power

$\int (C.V'.V) = C.V^2 = \int P.dt = E_{stored}$, Integrating by the time t, Power over time = Energy

To avoid confusion with electric field, $W_{done}$ or $W$ will be used instead of $E_{stored}$ and $E$ for energy.

$W=CV^2/2 = qV/2$