CHPTR 25 Notes: Electric Current, Resistance, and EMF

Electric Current

  • Definition of Electric Current (I)

    • Any motion of charge (positive or negative) from one place to another is defined as electric current.

    • The amount of charge, measured in coulombs, passing by every second is equal to the current in amperes.

    • Formula:
      I = \frac{dQ}{dt}

    • Thus, amperes can be defined as coulombs per second.

Microscale Behavior of Electrons

  • Microscopic Description of Current

    • When electrons move through conductors, their paths zigzag due to collisions with atoms.

    • In the absence of an external electric field, there is no net flow of electrons.

Applications of Electric Current

  • Types of Current Used

    • Common electric currents used in applications:

    • Radio and TV generally utilize currents measured in milliamperes (mA) and microamperes (μA).

    • Computer currents range from nanamperes (nA) to picoamperes (pA).

    • Speed of Electrons

    • The average speed of individual electrons is approximately 10^6 m/s.

    • Overall drift velocity of the electrons (the net velocity) is about 10^{-4} m/s.

Setup for Electric Current

  • Current Flow Mechanism

    • When a conductor is connected to a battery, it creates an electric field inside the conductor leading to the flow of current (electrons).

    • Conventional Current Direction

    • The direction of “conventional current” (I) is defined as the direction that positive charges would move.

Current Density

  • Definitions and Formulas for Current Density (J)

    • Current density relates the flow of charge with the number of charged particles:

    • dQ = qN = qnAv_ddt

    • Where:

      • n = number of charged particles per unit volume

      • A = cross-sectional area of conductor

      • v_d = drift velocity of electrons

    • Thus, current density can be expressed as:
      J = \frac{I}{A} = nqv_d

Resistivity

  • Understanding Resistivity

    • For most metals, the relationship between current density (J) and electric field (E) is approximately linear:

    • J \sim E

    • Definition of Resistivity ($\rho$)

    • Resistivity is defined as:
      \rho = \frac{E}{J}

    • Note: Resistivity is the reciprocal of conductivity.

    • Temperature Dependence of Resistivity

    • Resistivity varies with temperature according to the formula:
      \rho(T) = \rho0[1 + \alpha(T - T0)]

    • Where:

      • \alpha = temperature coefficient of resistivity

      • T_0 = reference temperature.

Conductivity and Resistance

  • Relationship Between Resistance (R) and Other Variables

    • For a conductor of length L, with resistance derived from resistivity:

    • Electric Potential difference formula:
      V = \rho \frac{I}{A} \cdot L

    • Consequently, the relationship between voltage and current can be expressed as:
      \frac{V}{I} = \frac{\rho L}{A} = R

    • Where:

      • R = resistance and is measured in ohms (Ω).

Resistance and Temperature

  • Resistance vs Temperature

    • The resistance of a metal varies approximately linearly with temperature similarly to resistivity:

    • R = R0 + \alpha (T - T0)

    • This means that as temperature increases, so does resistance.

Ohm's Law

  • Basic Concept

    • When a potential difference is applied across a conductor, an electric current is generated.

    • Ohm's Law

    • The ratio of voltage (V) to current (I) remains constant and is termed resistance (R).

    • Expressed as:
      V_{ab} = IR

Electromotive Force (EMF)

  • Definition of EMF ($\varepsilon$)

    • Measured in volts, which are equivalent to joules per coulomb.

    • EMF represents energy per unit charge, analogous to electric potential.

    • Example: An EMF of 9 volts means that 9 joules of work are done on each coulomb of charge that passes through it.

    • Sources of EMF

    • Common sources include: fuel cells, solar cells, generators, batteries, and thermocouples, all of which convert energy from some form into electric potential energy.

Terminal Voltage vs Ideal EMF

  • Terminal Voltage

    • The terminal voltage of a circuit is described by the equation:
      V_{ab} = \varepsilon - Ir

    • Where:

      • r = internal resistance of the battery.

  • Current in Circuit

    • The current in the circuit can be calculated with:
      I = \frac{\varepsilon}{R + r}

    • Where R is the total resistance in the circuit.

Electrical Energy and Power

  • Energy Conversion and Power Expression

    • Electrical energy delivered by the source gets converted into various forms such as heat, light, etc.

    • The power delivered by a battery to the load or resistor is given by:

    • General power formula is:
      P = VI

    • Or expressed with respect to resistance:
      P = I^2 R = \frac{V^2}{R}

Measuring Instruments

  • Voltmeter

    • A voltmeter is used to measure voltage and is connected in parallel to the component across which the voltage is measured.

    • An ideal voltmeter possesses infinite resistance to ensure that no current flows through it.

  • Ammeter

    • An ammeter measures current and is connected in series with other circuit components.

    • Ideally, an ammeter should have zero resistance to prevent alteration of the current being measured.

Resistors in Series (Chapter 26)

  • Behavior of Series Circuit

    • In a series circuit, the current flowing through each resistor remains the same.

    • The equivalent resistance of resistors in a series is the sum of the individual resistances:
      R{eq} = R1 + R_2 + …

    • The sum of voltages across each resistor is equal to the total voltage supplied by the source.

Resistors in Parallel (Chapter 26)

  • Behavior of Parallel Circuit

    • In a parallel arrangement, the potential difference (voltage) across each resistor is the same.

    • Equivalent Resistance in Parallel

    • The equivalent resistance of resistors connected in parallel is always less than the smallest individual resistance in the group:
      \frac{1}{R{eq}} = \frac{1}{R1} + \frac{1}{R_2} + …