Ohm's Law

Ohm’s law is one of the fundamental laws of electricity, and connects resistance to the voltage and current of an electrical device.

Definition §

Ohm's Law

$$V=IR$$

Note

• $V$ = Electrical Potential difference (i.e. voltage) (in $V$)
• $I$ = Current flowing through component (in $\text{A}$)
• $R$ = Resistance of the component. (in $\Omega$)

Derivation using Drude Model of Conduction §

Drude Model of Conduction states that electrons move in random directions, but the net movement can be achieved through the use of an Electric Field.

Using Newton’s second law of motion, and the definition of an electric field:

$$F=q\times \vec{E}=m\times \vec{a}\to \vec{a}=q\frac{\vec{E}}{m}$$

The acceleration of an electron, then, is given by:

$${\vec{a}}_e=\frac{\vec{E}{q}_e}{m_e}$$

Electrons in this field experience this acceleration, until they collide with another electron or particle. If we assume the average time between collisions is $\tau$, then

$$v_{drift}={\vec{a}}_e\times \tau =\frac{q_e\vec{E}\tau }{m_e}$$

We can connect this to the Current Density, $J$:

$$J=q_e{\rho }_e{v}_{drift}\to v_{drift}=\frac{J}{q_e{\rho }_e}$$

$$\frac{J}{q_e{\rho }_e}=\frac{q_e\vec{E}\tau }{m_e}$$

$$J=\frac{q_e^2\vec{E}\tau }{m_e}$$

Which shows that $J\propto \vec{E}$ (which was kind of obvious anyway, but still.)

Conductivity & Resistivity is defined to simplify this equation:

$$J=\sigma \vec{E}=\frac{\vec{E}}{\rho }$$

$J\propto \sigma \text{(conductivity)}\propto \frac{1}{\rho }(\frac{1}{\text{resistivity}})$

Finally, we can now expand this model up to a larger scale.

Assume we have a conducting line of cylinder of length $l$ and area $A$

The electric field can be represented as the gradient of the Electrical Potential difference over the length $l$:

$$\vec{E}=\frac{\Delta V}{l}$$

We can also now directly look at the current, since we have area:

$$J=\frac{I}{A}$$

And we define resistance to be $R=\frac{l}{A\sigma }$

Finally:

$$J=\frac{\vec{E}}{\rho }$$

\dfrac{I}{A} = \dfrac{\Delta V}{l}\sigma$$$$I = \dfrac{\Delta VA}{l}\sigma = \dfrac{\Delta V}{R}

So we get $\Delta V=IR$, proving Ohm’s law

Ohmic Vs Non-Ohmic §

Any device that obeys Ohm’s law, i.e. has a constant gradient for a voltage current graph is said to be ohmic. Conversely, any device that has a non constant gradient, i.e. doesn’t obey Ohm’s law, is a non-ohmic device.