Chapter 18: Electric Currents – Study Notes

Chapter 18: Electric Currents

The Electric Battery

The electric battery is a device that transforms chemical energy into electrical energy, providing a source of constant potential difference for electric circuits.

• Voltaic Cell: Discovered by Alessandro Volta, a simple electric cell consists of dissimilar metals connected by an electrolyte, which is a conductive solution.

• Operation: Chemical reactions within the cell create a potential difference between the terminals by slowly dissolving them. This potential difference is maintained until one terminal is completely dissolved.

• Battery Definition: Originally, a battery referred to several cells connected together, but now a single cell is often called a battery.

• Application: Batteries are used in portable electronics, vehicles, and many other devices.

Electric Current

Electric current is the rate at which electric charge flows through a conductor. It is a fundamental concept in understanding how electricity moves in circuits.

• Definition: Electric current ($I$) is defined as $I = \frac{\Delta Q}{\Delta t}$, where $\Delta Q$ is the charge passing through a point in time $\Delta t$.

• Unit: The unit of electric current is the ampere (A).

• Complete Circuit: Current can only flow in a complete circuit, which provides a path from one battery terminal, through the circuit, and back to the other terminal.

• Direction: By convention, current flows from positive (+) to negative (−) terminal. However, electrons actually flow in the opposite direction.

• Types of Current: Not all currents consist of electrons; for example, ionic currents in electrolytes.

Ohm’s Law: Resistance and Resistors

Ohm’s law describes the relationship between voltage, current, and resistance in a conductor. Resistance is a property that opposes the flow of electric current.

• Ohm’s Law: $V = IR$ (Equation 18-2), where $V$ is voltage, $I$ is current, and $R$ is resistance.

• Resistance: The ratio of voltage to current; measured in ohms ($\Omega$).

• Ohmic Materials: Materials with constant resistance, independent of voltage, are called ohmic. Nonohmic materials do not follow Ohm’s law.

• Resistors: Standard resistors are manufactured for use in circuits and are color-coded to indicate their value and precision.

• Clarifications:

  • Batteries maintain a nearly constant potential difference; current varies.

  • Resistance is a property of a material or device.

  • Current is not a vector but has a direction.

  • Current and charge are not used up; charge entering one end of a circuit exits the other end.

Resistivity

Resistivity is a material property that quantifies how strongly a material opposes the flow of electric current.

• Formula: $R = \rho \frac{l}{A}$ (Equation 18-3), where $R$ is resistance, $\rho$ is resistivity, $l$ is length, and $A$ is cross-sectional area.

• Material Dependence: $\rho$ is characteristic of the material.

• Temperature Dependence: For most materials, resistivity increases with temperature: $\rho = \rho_0 [1 + \alpha (T - T_0)]$ (Equation 18-4), where $\alpha$ is the temperature coefficient.

• Semiconductors: May have resistivities that decrease with temperature.

Table: Resistivity and Temperature Coefficients (at 20°C)

Material	Resistivity ($\rho$) [$\Omega \cdot m$]	Temperature Coefficient ($\alpha$) [1/°C]
Copper	1.68 × 10−8	0.0039
Aluminum	2.82 × 10−8	0.00429
Iron	9.71 × 10−8	0.00651
Additional info: Values inferred for common metals.

Electric Power

Electric power is the rate at which electrical energy is transferred or converted by a device.

• Definition: $P = IV$ (Equation 18-5), where $P$ is power, $I$ is current, and $V$ is voltage.

• Units: The unit of power is the watt (W).

• Ohmic Devices: For ohmic devices, $P = I^2 R$ (Equation 18-6a) or $P = \frac{V^2}{R}$ (Equation 18-6b).

• Energy Consumption: Electric bills measure energy in kilowatt-hours (kWh), not power.

• Conversion: $1 \text{ kWh } = 3.6 \times 10^6 \text{ J}$

Power in Household Circuits

Household circuits are designed to safely deliver electrical power. Safety devices prevent overheating and fire hazards.

• Wire Resistance: Household wires have low resistance, but high current can cause dangerous heating.

• Fuses: One-use safety devices that break the circuit if current exceeds a safe value.

• Circuit Breakers: Reusable switches that open the circuit when current is too high; can be reset.

• Application: Used to prevent electrical fires and protect appliances.

Alternating Current (AC)

Alternating current is a type of electric current in which the direction and magnitude vary periodically, typically in a sinusoidal manner.

• Direct Current (DC): Flows steadily in one direction (from batteries).

• Alternating Current (AC): Varies sinusoidally (from power plants).

• Voltage and Current: $V = V_0 \sin(\omega t)$ (Equation 18-7a), $I = I_0 \sin(\omega t)$ (Equation 18-7b).

• Power: Instantaneous power is $P = IV$.

• Average Power: For AC, average power is $P_{avg} = I_{rms} V_{rms}$.

• Root Mean Square (rms) Values:

  • $I_{rms} = \frac{I_0}{\sqrt{2}}$ (Equation 18-8a)

  • $V_{rms} = \frac{V_0}{\sqrt{2}}$ (Equation 18-8b)

Microscopic View of Electric Current

At the microscopic level, electric current is due to the movement of charge carriers, typically electrons, within a conductor.

• Electron Motion: Electrons have random thermal velocities due to temperature.

• Drift Velocity: When a potential difference is applied, electrons acquire an average drift velocity ($v_d$), much smaller than their thermal speed.

• Current Relation: $I = n q v_d A$ (Equation 18-10), where $n$ is the number of charge carriers per unit volume, $q$ is the charge of each carrier, $A$ is the cross-sectional area.

Superconductivity

Superconductivity is a phenomenon where certain materials exhibit zero electrical resistivity below a critical temperature.

• Temperature Dependence: In general, resistivity decreases as temperature decreases.

• Critical Temperature: At a specific low temperature, resistivity drops abruptly to zero.

• Persistent Currents: Currents can flow for years without decreasing, even without a potential difference.

• High-Temperature Superconductors: Recent discoveries have found materials with critical temperatures up to 160 K.

• Applications: Used in MRI machines, maglev trains, and advanced electronics.

Electrical Conduction in the Human Nervous System

The human nervous system relies on the flow of electric charge to transmit signals between cells.

• Neurons: Basic elements of the nervous system, consisting of a cell body, dendrites, and an axon.

• Signal Transmission: Signals are received by dendrites, propagated along the axon, and transmitted through synapses.

• Membrane Potential: Depends on a dipole layer of charge and different ion concentrations inside and outside the cell.

• Action Potential: Neurons respond to stimuli and conduct electrical signals in the form of action potentials.

• Propagation: The action potential travels along the axon membrane, enabling communication throughout the body.

• Application: Understanding electrical conduction in neurons is fundamental to neuroscience and medical diagnostics.

Summary Table: Key Equations and Concepts

Concept	Equation	Unit
Electric Current	$I = \frac{\Delta Q}{\Delta t}$	Ampere (A)
Ohm's Law	$V = IR$	Volt (V), Ohm ($\Omega$)
Resistance	$R = \rho \frac{l}{A}$	Ohm ($\Omega$)
Electric Power	$P = IV$	Watt (W)
AC rms Values	$I_{rms} = \frac{I_0}{\sqrt{2}}$, $V_{rms} = \frac{V_0}{\sqrt{2}}$	Ampere (A), Volt (V)
Microscopic Current	$I = n q v_d A$	Ampere (A)
Additional info: Table summarizes main equations for quick reference.