Resistance and Ohm's law Knowledge-Based Study

Study notes

1. Electric Current and Potential Difference

   In direct-current (DC) circuits, electric current (I) represents the rate at which electric charge flows past a cross-section of a conductor. In the International System of Units (SI), charge is measured in coulombs (C) and time in seconds (s), making the unit of current the ampere (A), where 1 A = 1 C/s.

   For charges to flow continuously through a closed loop, an electric potential difference (V), commonly called voltage, must be established across the circuit. Voltage is defined as the work done per unit charge and is measured in volts (V), where 1 V = 1 J/C. A potential difference acts as the electrical driving influence that propels charge carriers through a conducting path.

   Electrical Resistance

   As electric charges move through a conductor, they undergo frequent collisions with the lattice ions of the material. This opposition to the flow of electric charge is termed electrical resistance (R), measured in ohms ($\Omega$).

   For a uniform conductor of length L and cross-sectional area A, resistance depends directly on its physical dimensions and its material composition:

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

   • Resistivity ($\rho$): An intrinsic property of the material measured in ohm-meters ($\Omega \cdot \text{m}$). Good conductors (such as copper and aluminum) possess very low resistivity, whereas insulators have extremely high resistivity.
   • Length (L): Resistance is directly proportional to conductor length. Longer wires present more opportunities for scattering events.
   • Cross-sectional Area (A): Resistance is inversely proportional to cross-sectional area. A thicker wire provides a wider path, lowering total resistance.

   Ohm's Law and Ohmic Conductors

   Ohm's law states that for many conducting materials kept at a constant temperature, the electric current passing through the conductor is directly proportional to the potential difference applied across it:

   $$V = I R$$

   This relationship can be rearranged to calculate any one of the three fundamental variables when the other two are known:

   • $I = \frac{V}{R}$ (Current calculation)
   • $R = \frac{V}{I}$ (Resistance calculation)
   • $V = I R$ (Voltage calculation)

   Materials that strictly obey this linear relationship are called ohmic conductors. A plot of voltage versus current for an ohmic component is a straight line passing through the origin, where the slope of the line equals the resistance R. Components whose resistance changes as current or temperature varies (such as diodes or filament bulbs) are classified as non-ohmic.

   Worked Example and Circuit Analysis

   Consider a simple DC circuit consisting of a constant 12 V power supply connected across an ohmic resistor of 4.0 $\Omega$.

   Step 1: Determine the Circuit Current

   Using Ohm's law:

   $$I = \frac{V}{R} = \frac{12\text{ V}}{4.0\,\Omega} = 3.0\text{ A}$$

   Step 2: Effect of Modifying Conductor Dimensions

   If the original resistor wire is replaced by a wire of the identical material and cross-sectional area, but with twice the length ($2L$), its new resistance becomes:

   $$R' = \rho \frac{2L}{A} = 2R = 8.0\,\Omega$$

   Under the same 12 V potential difference, the resulting current is halved:

   $$I' = \frac{12\text{ V}}{8.0\,\Omega} = 1.5\text{ A}$$

   This demonstrates the inverse relationship between resistance and current when source potential is held constant.