Resistance and Ohm's Law

When voltage is applied across a material, current flows. However, the material itself opposes this flow of current. This opposition is called resistance. Understanding resistance and its relationship with voltage and current forms the basis of all circuit analysis.

What is Resistance?

Resistance ($R$) is a measure of how much a material restricts the flow of electrical current. It is analogous to the friction in a pipe that restricts the flow of water.

The unit of resistance is the Ohm ($\Omega$).

Resistivity and Physical Geometry

The resistance of a specific component (like a wire) depends on both what it is made of and its physical dimensions.

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

where:

• $R$ = resistance [$\Omega$]
• $\rho$ (rho) = resistivity of the material [$\Omega\cdot\text{m}$]
• $L$ = length of the material [m]
• $A$ = cross-sectional area [m²]

Resistivity ($\rho$) is a fundamental material property. Conductors like copper have a very low resistivity, while insulators like rubber have a very high resistivity.

Engineering Meaning: If you double the length of a wire, you double its resistance. If you double its cross-sectional area (make it thicker), you halve its resistance.

Ohm's Law

The foundational relationship between voltage, current, and resistance was discovered by Georg Ohm. Ohm's Law states that the voltage across a conductor is directly proportional to the current flowing through it, provided all physical conditions and temperatures remain constant.

The mathematical formulation is:

$V = I R$

where:

• $V$ = voltage (potential difference) [V]
• $I$ = current [A]
• $R$ = resistance [$\Omega$]

Using Ohm's Law

Ohm's Law can be rearranged depending on which variable you need to find:

• To find Voltage: $V = I \cdot R$
• To find Current: $I = \frac{V}{R}$
• To find Resistance: $R = \frac{V}{I}$

Worked Example

Problem: A copper wire has a resistance of $0.5 \, \Omega$. It is connected to a 12 V battery. What is the current flowing through the wire? If the wire's length is 10 m and its cross-sectional area is $0.34 \text{ mm}^2$, what is the resistivity of the copper?

Solution:

Step 1 — Calculate the Current using Ohm's Law: $I = \frac{V}{R} = \frac{12 \text{ V}}{0.5 \, \Omega} = 24 \text{ A}$

Step 2 — Calculate the Resistivity: First, ensure all units are in standard SI (convert mm² to m²): $A = 0.34 \text{ mm}^2 = 0.34 \times 10^{-6} \text{ m}^2$

Rearrange the resistance formula to solve for $\rho$: $R = \rho \frac{L}{A} \implies \rho = \frac{R \cdot A}{L}$

$\rho = \frac{0.5 \times (0.34 \times 10^{-6})}{10} = 1.7 \times 10^{-8} \, \Omega\cdot\text{m}$

Engineering Check: $1.7 \times 10^{-8} \, \Omega\cdot\text{m}$ is the standard accepted value for the resistivity of copper at room temperature, confirming the calculation is physically sound.

Common Mistakes

1. Assuming Ohm's Law applies to everything: Ohm's law only applies strictly to "Ohmic" materials where resistance is constant regardless of applied voltage. Non-Ohmic devices, like diodes or lightbulbs (where resistance changes with temperature), do not have a linear $V-I$ relationship.
2. Ignoring unit conversions in the resistivity equation: Cross-sectional areas of wires are almost always given in mm² or AWG. They must be converted to m² before being used in $R = \rho L/A$.