Physics II: Current and Resistance

Faculty of Sciences Dept. of Physical & Chemical Sciences

  • BSB212, Physics II

  • Current and Resistance

  • O. Chimidza

  • Office: 302/109

2.1.1 The Electric Current

  • Definition: Electric current, or simply current, is the rate of flow of electric charge through a surface.

  • Key Points:   - Current is indicated by charges moving through a cross-sectional area (e.g., within a wire).   - If $$Q is the charge passing through that area in a time interval $$t, then:     - Average current, I_{av} = rac{Q}{ t}   - If the flow of charge is constant over time, the instantaneous current II is defined as:     - I=racdQdtI = rac{dQ}{dt}

  • SI Unit of Current: Ampere (A)   - $1A = 1C/s$ (one coulomb of charge passing a point in a circuit per second).   - Common subunits: milliampere (mA), microampere (μA).

  • Conventional Direction: The direction of current is set to the flow of positive charges, in contrast to the actual flow of negatively charged electrons in conductors like copper.

  • Mobile Charge Carriers: In metals, the charge carriers are electrons.

  • Current Density (J): The current per unit area and is given by:   - J=racIA=nqvdJ = rac{I}{A} = nqv_d   - where:     - $J$ = current density (A/m²)     - $n$ = number of charge carriers per unit volume (m³)     - $q$ = charge of each carrier (coulombs)     - $v_d$ = drift velocity of charge carriers (m/s)

  • Current in a Conductor:   - Total charge in a volume element of the conductor is:     - Q = nqA x   - If charge carriers move with drift speed $v_d$, the total current is given by:     - I = rac{Q}{ t} = nqv_dA

Example: Current in a Circuit

  • Given: A steady current of 2.5 A for 4.0 minutes:   - (a) Total charge passed:     - I=2.5A=2.5C/sI = 2.5A = 2.5C/s     -  t = 240s     - Q = I t = 2.5C/s imes 240s = 600C   - (b) Number of electrons:     - Q = Ne (charge per electron is approximately $1.602 imes 10^{-19}C$)     - N = rac{Q}{e} = rac{600C}{1.602 imes 10^{-19}C} ightarrow 3.8 imes 10^{21} electrons   - (c) Current calculation for 1 million electrons per second:     - Q = Ne = 10^6 imes (1.602 imes 10^{-19})C = 1.602 imes 10^{-13}C     - I = rac{Q}{ t} = rac{1.602 imes 10^{-13}C}{1s} = 0.1602A

2.1.2 Resistance and Ohm’s Law

  • Objective: Understand how charges move in conductors to produce current.

  • Electric Field: In a conductor, an electric field creates a force that causes charge to move, resulting in a current.

  • Current Density Equation:   - J=racIA=nqvdJ = rac{I}{A} = nqv_d

  • Ohm's Law:   - States the linear relationship between current density JJ and electric field EE:     - J=extσEJ = ext{σ}E   - $ ext{σ}$ = conductivity of the material (S/m), which depends on material properties and temperature.

  • For Ohmic Materials:   - JextisproportionaltoEJ ext{ is proportional to } E (Ohmic materials, e.g., metals)   - Nonohmic materials have nonlinear relationships.

Resistance of Conductors

  • Definition: Resistance $R$ is defined as:   - R=racVIR = rac{V}{I}

  • Geometric and Material Dependence:   - R=racholAR = rac{ ho l}{A} where:     - $ ho$ = resistivity (Ω·m)     - $l$ = length of the conductor (m)     - $A$ = cross-sectional area (m²)

  • Resistivity: Refers to the intrinsic resistance of a material to current flow, influenced by temperature and material properties.

  • Ohmic vs Non-Ohmic Behavior:   - Ohmic materials maintain a linear voltage-current characteristic;   - Non-Ohmic materials do not (e.g., diodes, transistors).

Resistance Examples

  1. Circuits:    - Understanding resistance in varied copper conductors at the same temperature based on geometry.

  2. Calculating Resistance of a Bulb:    - If a flashlight bulb draws 300 mA from a 1.5 V source, resistance can be calculated as:
       R=racVI=rac1.5V0.3Aightarrow5ΩR = rac{V}{I} = rac{1.5V}{0.3A} ightarrow 5Ω

  3. Example Solutions:    - The resistance varies according to shape by manipulating cross-sectional area and length.

2.1.3 Electrical Energy and Power

  • Concept: An electric circuit involving a battery connected to a resistor facilitates power transfer and energy consumption.

  • Energy in Circuits:   - Electric potential energy is converted to thermal energy across resistors by potential difference ($V$).

  • Power Loss in Resistors:   - Power PP is derived from the potential difference VV and current II as follows:     - P=IV=I2R=racV2RP = IV = I^2R = rac{V^2}{R} (formulas demonstrating the equivalence of these expressions based on Ohm’s Law)

  • Units: Power is measured in watts (W) equivalent to Joules per second (J/s).

Example: Power Consumption

  • Electric Heater: An electric heater draws 110 V across a resistance of 8 Ω:   - Current: I=racVR=rac110V8Ωightarrow13.75AI = rac{V}{R} = rac{110V}{8Ω} ightarrow 13.75A   - Power: P=I2R=13.75A2imes8Ωightarrow1.52WP = I^2 R = 13.75A^2 imes 8Ω ightarrow 1.52W

2.1.4 Energy Conversion in Household Circuits

  • Energy Units: Power companies bill energy usage in kilowatt-hours (kWh).   - 1 kWh = 1 kW consumed over 1 hour = 3.6 x 10^6 J.

  • Cost Calculation Examples:   - How to compute energy cost for different consumption scenarios based on device specifics.

Example Calculations:

  1. Lightbulb Cost Calculation for 24 h at P8.00 per kWh:    - P=100W=0.1kWP = 100W = 0.1 kW
       - Energy=Pimest=0.1kWimes24h=2.4kWhEnergy = P imes t = 0.1 kW imes 24h = 2.4 kWh    - Cost=Energyimesrate=2.4kWhimesP8.00=P19.20Cost = Energy imes rate = 2.4 kWh imes P8.00 = P19.20

  2. Total Current for Appliances in Parallel:    - Determining total current from multiple devices operating simultaneously with their rated power to obtain the current drawn from the source.    - Issues concerning devices drawing excessive current vs rated cord limits leading to overheating dangers.