Question

Explain the factors that affect the heat generated in a current-carrying conductor. Derive its formula.

Answer 1

i. The heat (H) generated in a conductor depends on three main factors:

1. Electric Current (I): The heat produced is directly proportional to the square of the current passing through the conductor.

H ∝ I²

2. Resistance (R): The heat produced is directly proportional to the electrical resistance of the conductor.

H ∝ R

3. Time (t): The heat produced is directly proportional to the time duration for which the current flows.

H ∝ t

Combining all three factors gives Joule's Law of Heating:

H ∝ I²Rt

or

H = I²Rt

ii. Derivation of the Formula:

Let a current (I) flow through a conductor of resistance (R) when a potential difference (V) is applied across its ends for a time (t).

1. Work Done by Charge: By definition of potential difference, the work done (W) to move a charge (Q) across the conductor is:

W = V × Q  ....(1)

2. Expressing Charge in Terms of Current: Since electric current is the rate of flow of charge (\(I = \frac{Q}{t}\)), we can write:

Q = I × t   .......(2)

3. Substituting Equation 2 into Equation 1:

W = V × I × t

4. Applying Ohm's Law: According to Ohm's Law, (V = I × R). Substituting this value of (V):

W = (I × R) × I × t

W = I²Rt

5. Conversion to Heat: Assuming all the electrical work done by the current is converted entirely into heat energy (H = W):

H = I²Rt