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 $

Table of Contents