Mathematical Derivation of the Heating Effect of Electric Current

The heating effect of electric current refers to the phenomenon where electric current passing through a conductor produces heat. This effect is utilized in various electrical appliances such as heaters, toasters, and electric stoves. The mathematical derivation of the heating effect can be done using Ohm's Law and the concept of electrical power.

Ohm's Law:
Ohm's Law states that the current flowing through a conductor is directly proportional to the voltage applied across it and inversely proportional to its resistance. Mathematically, Ohm's Law can be expressed as:

V = I * R

Where:
V is the voltage applied across the conductor,
I is the current flowing through the conductor, and
R is the resistance of the conductor.

Electrical Power:
The power dissipated in a circuit can be calculated using the formula:

P = V * I

Where:
P is the power dissipated,
V is the voltage across the circuit, and
I is the current flowing through the circuit.

Derivation:
To derive the heating effect of electric current, we need to calculate the power dissipated in a conductor.

1. Consider a conductor with resistance R through which a current I is flowing.

2. According to Ohm's Law, the voltage across the conductor can be expressed as:
V = I * R

3. Substituting this value of V in the power formula, we get:
P = (I * R) * I
= I^2 * R

4. This equation shows that the power dissipated in a conductor is proportional to the square of the current flowing through it and the resistance of the conductor.

5. As power is directly related to heat, the heating effect of electric current is also proportional to the square of the current and the resistance.

6. This derivation mathematically explains why the heating effect is more prominent in high-resistance conductors and when the current flowing through them is high.

In conclusion, the heating effect of electric current can be derived mathematically by applying Ohm's Law and the concept of electrical power. The power dissipated in a conductor is given by P = I^2 * R, where P is the power, I is the current, and R is the resistance. This equation demonstrates that the heating effect is directly proportional to the square of the current and the resistance.