Derivation of Ohm's Law

Introduction:
Ohm's law is a fundamental principle in physics that describes the relationship between current, voltage, and resistance in an electric circuit. It states that the current passing through a conductor is directly proportional to the voltage applied across it, and inversely proportional to the resistance of the conductor. The mathematical expression of Ohm's law is given by the equation: V = IR, where V is the voltage, I is the current, and R is the resistance.

Derivation:
To derive Ohm's law, we consider a simple circuit consisting of a power source (battery), a resistor, and connecting wires.

1. Assumptions:
- The resistor is an ohmic conductor, meaning its resistance remains constant regardless of the applied voltage or current.
- The connecting wires have negligible resistance.

2. Definition of Voltage and Current:
- Voltage (V) is defined as the electrical potential difference between two points in a circuit. It is measured in volts (V).
- Current (I) is defined as the rate of flow of electric charge through a conductor. It is measured in amperes (A).

3. Ohm's Law:
- According to Ohm's law, the current flowing through a resistor is directly proportional to the voltage applied across it and inversely proportional to its resistance.
- Mathematically, this can be expressed as V ∝ I or V = kI, where k is a constant of proportionality.
- Additionally, we know that the resistance (R) of a conductor can be represented as R = V/I.

4. Combining Equations:
- By substituting the expression for resistance (R) in the equation V = kI, we get V = (V/I)I.
- Simplifying, we find V = V, which is a tautology.
- This implies that the constant of proportionality k is equal to 1, and the equation becomes V = I.

5. Conclusion:
- Therefore, we have derived Ohm's law, which states that the voltage across a resistor is equal to the current flowing through it multiplied by its resistance: V = IR.

Summary:
Ohm's law states that the voltage across a resistor is directly proportional to the current flowing through it and inversely proportional to its resistance. The mathematical expression of Ohm's law is V = IR.