Introduction

This chapter explores the fundamental concepts of electric current, which is the flow of electric charge. It builds upon the understanding of electric fields and potential difference from previous chapters. We examine steady currents in conductors, inspired by everyday devices like torches and clocks, and contrast them with transient phenomena like lightning. The chapter lays the groundwork for understanding electrical circuits and their behavior.

Electric Current

Electric current ($I$) is defined as the net rate of flow of electric charge across a given surface. For a steady current, it is the net charge ($q$) passing through the surface per unit time ($t$), $I = q/t$. More generally, for non-steady currents, it is the limit of the ratio of charge change ($\Delta Q$) to time interval ($\Delta t$) as $\Delta t$ approaches zero: $I = \lim_{\Delta t \to 0} \frac{\Delta Q}{\Delta t}$. The SI unit of electric current is the Ampere (A). Currents can range from microamperes in nerves to tens of thousands of amperes in lightning.

Electric Currents In Conductors

Electric currents are established in conductors when free charge carriers, such as electrons in metals or ions in electrolytes, are set in motion by an applied electric field. In metals, electrons move due to thermal agitation and are accelerated by an electric field. However, collisions with fixed ions randomize their direction, resulting in a net drift velocity superimposed on their random motion. This drift velocity is responsible for the continuous flow of charge, i.e., the electric current.

A steady electric field, typically maintained by a source like a cell or battery, is necessary for a continuous current. Without such a source, any initial current caused by applied charges would quickly cease as the charges neutralize.

Ohm’s Law

Ohm's Law states that for a conductor at a constant temperature, the potential difference ($V$) across its ends is directly proportional to the current ($I$) flowing through it:

$V = IR$

where $R$ is the resistance of the conductor, a constant of proportionality. The SI unit of resistance is the Ohm ($\Omega$). The resistance of a conductor is directly proportional to its length ($l$) and inversely proportional to its cross-sectional area ($A$):

$R = \rho \frac{l}{A}$

where $\rho$ is the resistivity of the material, a fundamental property that depends on the material and temperature. In terms of electric field ($E$) and current density ($j$), Ohm's Law can be expressed as $E = \rho j$ or $j = \sigma E$, where $\sigma = 1/\rho$ is the conductivity.

Example 3.1: Estimates the drift speed of conduction electrons in a copper wire, comparing it to thermal speeds and the speed of electric field propagation.

Example 3.2: Addresses conceptual questions about current establishment, drift velocity, and the role of electron density.

Example 3.3: Calculates the steady temperature of a nichrome heating element based on its resistance change with temperature and the applied voltage and current.

Example 3.4: Determines the temperature of a bath using a platinum resistance thermometer and the given resistance values at different temperatures.

Drift Of Electrons And The Origin Of Resistivity

In the presence of an electric field ($E$), electrons in a conductor experience a force ($F = -eE$) and accelerate. However, due to frequent collisions with the lattice ions, they acquire a constant average velocity called the drift velocity ($v_d$). This drift velocity is proportional to the electric field and the average time ($\tau$) between collisions:

$v_d = -\frac{eE}{m} \tau$

where $e$ is the magnitude of electron charge and $m$ is its mass. The current density ($j$) is related to the drift velocity by $j = nev_d$, where $n$ is the number density of free electrons. This leads to Ohm's Law in microscopic form, $j = \sigma E$, with conductivity $\sigma = \frac{ne^2\tau}{m}$.

Mobility ($\mu$) is defined as the magnitude of drift velocity per unit electric field: $\mu = |v_d|/E = \frac{e\tau}{m}$.

Limitations Of Ohm’s Law

Ohm's Law, $V=IR$, is not universally applicable. Deviations occur when:

• The proportionality between $V$ and $I$ breaks down (e.g., in semiconductors at high currents).
• The relationship between $V$ and $I$ depends on the sign of $V$ (e.g., in diodes).
• The $V-I$ relation is not unique (e.g., in materials like GaAs).

Materials and devices that do not obey Ohm's Law are called non-ohmic.

Figures 3.5, 3.6, and 3.7 illustrate these deviations with characteristic curves.

Resistivity Of Various Materials

Materials are classified based on their resistivity ($\rho$):

• Conductors: Low resistivity ($10^{-8}$ to $10^{-6} \Omega \text{m}$), due to a large number of free charge carriers.
• Semiconductors: Intermediate resistivity, with values that typically decrease with increasing temperature.
• Insulators: Very high resistivity ($> 10^{10} \Omega \text{m}$), with negligible free charge carriers.

The ability to control the resistivity of semiconductors by doping makes them crucial for electronic devices.

Temperature Dependence Of Resistivity

The resistivity of most metallic conductors increases with temperature, approximately linearly over a limited range:

$\rho_T = \rho_0 [1 + \alpha (T - T_0)]$

where $\alpha$ is the temperature coefficient of resistivity. Alloys like Nichrome, Manganin, and Constantan have low temperature coefficients, making them suitable for precision resistors.

For semiconductors, resistivity generally decreases with increasing temperature because the number of charge carriers increases significantly with temperature, outweighing the effect of increased collision frequency.

Figure 3.8, 3.9, and 3.10 show the temperature dependence of resistivity for different materials.

Electrical Energy, Power

When a current $I$ flows through a conductor of resistance $R$ across which a potential difference $V$ is maintained, electrical energy is dissipated, usually as heat. The rate at which this energy is dissipated is the power ($P$).

$P = VI = I^2R = \frac{V^2}{R}$

This power comes from the source (e.g., a cell) which does work to maintain the current. For transmission of electrical power over long distances, the voltage is stepped up to very high values to minimize power loss ($I^2R_{cable}$) in the transmission wires, due to the inverse square relationship between power loss and voltage ($P_{loss} \propto 1/V^2$ if power delivered is constant).

Cells, Emf, Internal Resistance

A cell is a source of electrical energy that maintains a steady current in a circuit. It consists of two electrodes immersed in an electrolyte. The electromotive force (emf, $\epsilon$) of a cell is the work done per unit charge by the cell in moving charge from its negative to its positive terminal, which is equal to the potential difference between the terminals in an open circuit (no current flowing).

When a current $I$ flows through the cell, the terminal voltage $V$ across the cell is less than its emf due to the potential drop across its internal resistance ($r$):

$V = \epsilon - Ir$

For a circuit with external resistance $R$ connected to a cell, the current is $I = \frac{\epsilon}{R+r}$. The maximum current from a cell occurs when $R=0$ ($I_{max} = \epsilon/r$), but this can damage the cell.

Figure 3.12 illustrates a cell and its symbol.

Cells In Series And In Parallel

Cells In Series

When cells are connected in series (positive terminal of one to the negative terminal of the next), their emfs add up, and their internal resistances also add up:

$\epsilon_{eq} = \epsilon_1 + \epsilon_2 + \ldots + \epsilon_n$

$r_{eq} = r_1 + r_2 + \ldots + r_n$

If the polarity is reversed for some cells, their emfs are subtracted.

Figure 3.13 shows two cells in series.

Cells In Parallel

When cells with the same emf are connected in parallel (positive terminals together, negative terminals together), the equivalent emf is the same as that of a single cell, and the equivalent internal resistance is less than the smallest individual internal resistance:

$\epsilon_{eq} = \epsilon$ (if all $\epsilon_i$ are equal)

$\frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2} + \ldots + \frac{1}{r_n}$

If the emfs are different, the equivalent emf and resistance are calculated using weighted averages based on internal resistances.

Figure 3.14 illustrates two cells in parallel.

Kirchhoff’s Rules

Kirchhoff's rules are used to analyze complex electrical circuits:

1. Junction Rule (First Rule): The algebraic sum of currents entering any junction in a circuit is zero. This is based on the conservation of charge. ($\sum I_{in} = \sum I_{out}$)
2. Loop Rule (Second Rule): The algebraic sum of potential changes (voltages) around any closed loop in a circuit is zero. This is based on the conservation of energy.

Figure 3.15 illustrates these rules with an example circuit. Example 3.5 applies Kirchhoff's rules and symmetry to find the equivalent resistance and currents in a cube network. Example 3.6 demonstrates the application of Kirchhoff's rules to a complex network to find currents in each branch.

Wheatstone Bridge

A Wheatstone bridge is a circuit used for measuring an unknown resistance. It consists of four resistors ($R_1, R_2, R_3, R_4$) arranged in a bridge formation, with a galvanometer connected across one diagonal and a voltage source across the other. The bridge is balanced when the galvanometer shows zero current, which occurs when the ratio of resistances in the arms is equal:

$\frac{R_1}{R_2} = \frac{R_3}{R_4}$

This balance condition allows for precise determination of an unknown resistance by adjusting one of the known resistances until the galvanometer reads zero.

Figure 3.18 shows a typical Wheatstone bridge circuit. Example 3.7 calculates the current through the galvanometer in an unbalanced Wheatstone bridge.

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