Derive the expression for the heat produced due to a current 'I' flowing for a time interval 't' through a resistor 'R' having a potential difference 'V' across its ends. With which name is the relation known ? How much heat will an instrument of 12W produce in one minute if it is connected to a battery of 12V ? Or Explain with the help of a labelled circuit diagram how you will find the resistance of a combination of three resistors, of resistance $R_1$, $R_2$ and $R_3$, joined in parallel. Also mention how you will connect the ammeter and the voltmeter in the circuit when measuring the current in the circuit and the potential difference across one of the three resistors of the combination.

Part A: Derivation of Heat Produced

According to the text, the electric potential difference $V$ between two points is the work done $W$ to move a unit charge $Q$ (Section 11.2):

$$V = \frac{W}{Q} \implies W = V \times Q \quad \text{---(i)}$$

From the definition of electric current $I$ (Section 11.1):

$$I = \frac{Q}{t} \implies Q = I \times t \quad \text{---(ii)}$$

Substituting (ii) into (i), the work done by the source in time $t$ is:

$$W = V \times I \times t$$

Assuming all the electrical energy consumed is converted into heat energy ($H$):

$$H = VIt$$

Applying Ohm’s Law $V = IR$ (Section 11.4, Eq. 11.5):

$$H = (IR) \times I \times t$$

$$H = I^2Rt$$

This relation is known as Joule's Law of Heating.

Part B: Numerical Solution

Given:

Power $P = 12\text{ W}$

Time $t = 1\text{ minute} = 60\text{ s}$

Voltage $V = 12\text{ V}$

Since $P = VI$ and $H = VIt$, we can use $H = P \times t$:

$$H = 12\text{ W} \times 60\text{ s} = 720\text{ J}$$

---

OR: Resistors in Parallel

When three resistors $R_1, R_2,$ and $R_3$ are connected in parallel, the potential difference $V$ across each resistor remains the same (equal to the voltage of the battery). However, the total current $I$ is divided among the branches:

$$I = I_1 + I_2 + I_3$$

According to Ohm's Law ($I = V/R$):

$$\frac{V}{R_p} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}$$

Dividing by $V$ on both sides:

$$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$$

Instrument Connections:

1. Ammeter: Always connected in series in the circuit to measure the total current (Section 11.1).
2. Voltmeter: Always connected in parallel across the resistor or combination to measure the potential difference (Section 11.2).