Electric Current Ohms Law

3.1 Introduction

In Chapter 1, all the charges (positive or negative) we considered to be at rest. Charges in motion constitute an electric current. Lightning is one such phenomenon in which charges flow from the clouds to the earth through the atmosphere, sometimes with disastrous results. The flow of charges in lightning is not steady, but in our everyday life we use many devices where charges flow in a steady manner — such as cells powering torches, mains power lighting our homes, and currents that drive trains, fans and computers. Current Electricity, the topic of this chapter, deals with the steady flow of charges through electric circuits.

3.2 Electric Current

Imagine a small area held normal to the direction of flow of charges. Both the positive and the negative charges may flow forward and backward across the area. In a given time interval \(t\), let \(q_+\) be the net amount (i.e., forward minus backward) of positive charge flowing in the forward direction across the area. Similarly, let \(q_-\) be the net amount of negative charge flowing across the area in the forward direction. The net amount of charge flowing across the area in the forward direction in the time interval \(t\) is \(q = q_+ - q_-\). This is proportional to \(t\) for steady current flow.

The current is defined as the rate of flow of charge:

3.3 Electric Currents in Conductors

An electric charge experiences a force when an electric field is applied. If it is free to move, it will thus move contributing to a current. In nature, free charged particles do exist: like in upper layers of atmosphere called the ionosphere. However, in atoms and molecules, the negatively charged electrons and the positively charged nuclei are bound to each other and are thus not free to move. Bulk matter is made up of many molecules — a gram of water has about \(10^{22}\) molecules. These molecules are so closely packed that the electrons are no longer attached to individual nuclei. In some materials the electrons are still bound (insulators), but in conductors a few electrons are free to move within the bulk material. These materials, generally metals, are called conductors and develop a current when an electric field is applied.

If we consider a conductor with two ends maintained at different electric potentials, an electric field exists inside it which sets up the steady drift of electrons across the conductor — and so a steady current.

3.4 Ohm's Law

A basic law regarding flow of currents was discovered by G.S. Ohm in 1828, long before the physical mechanism responsible for flow of currents was discovered. Imagine a conductor through which a current \(I\) is flowing and let \(V\) be the potential difference between the ends of the conductor. Then Ohm's law states that

3.4.1 Resistance and Geometry

The resistance of a conductor depends on its length \(\ell\) and the cross-sectional area \(A\). Two identical conductors joined end-to-end (length \(2\ell\), same area) double the resistance: \(R \propto \ell\). Two identical conductors joined side-by-side (same length, area \(2A\)) halve the resistance: \(R \propto 1/A\). Combining,

where \(\rho\) is the resistivity of the material. Its reciprocal \(\sigma = 1/\rho\) is called the conductivity (SI unit: S/m or Ω⁻¹m⁻¹).

3.5 Drift of Electrons & Origin of Resistivity

As remarked before, an electron will suffer collisions with the heavy fixed ions, but after collision, it will emerge with the same speed as before. However, the direction of its velocity after the collision is completely random. So at a given time there is no preferential direction for the velocities of the electrons — the average velocity is zero. There is no net current.

When an external electric field \(\vec E\) is applied across the conductor, the electrons experience a force \(-e\vec E\). Between two successive collisions (average time \(\tau\), called relaxation time), each electron acquires an additional velocity \(\vec v = -\dfrac{e\vec E}{m}\tau\). The average over many electrons gives the drift velocity:

The negative sign tells us drift velocity is opposite to the applied field. Its magnitude is small (~10⁻⁴ m/s), yet it is enough to give measurable currents because the number density of electrons is huge.

3.5.1 Current Density and Microscopic Ohm's Law

Consider a conductor of cross-section \(A\) carrying current \(I\). If \(n\) is the free-electron density, then in time \(\Delta t\) the electrons travel a distance \(v_d\Delta t\), and the number of electrons crossing a section is \(n A v_d \Delta t\). Hence,

Substituting \(v_d = eE\tau/m\):

Thus resistivity arises from collisions of electrons with lattice ions. The smaller the relaxation time, the higher the resistivity.

3.5.2 Mobility

The mobility of a charge carrier is defined as the magnitude of drift velocity per unit electric field:

Resistivity of Common Materials (at 0 °C)

3.6 V–I Characteristics — Ohmic & Non-Ohmic

If we plot V (x-axis) against I (y-axis) for a metallic conductor at constant temperature, we get a straight line through the origin. Such conductors are called ohmic. Many devices, however, do not follow Ohm's law:

• Junction diode: conducts strongly only in one direction. V–I is non-linear and not symmetric about the origin.
• Gallium arsenide (GaAs): shows a negative resistance region (current decreases as V increases beyond a threshold).
• Thermistor / filament lamp: resistance changes strongly with temperature, giving curved V–I characteristics.