Two resistors, R₁ = 6 Ω and R₂ = 12 Ω, are connected in parallel to a 24 V battery. The circuit operates for 5 minutes.

(a) Calculate the total heat generated in both resistors.

(b) If each resistor has a power rating of 100 W, determine whether it is safe to use these resistors in the circuit.

Given,

• R₁ = 6 Ω
• R₂ = 12 Ω
• Battery voltage ($\text V$) = 24 V
• Operating time of the circuit ($\text t$) = 5 minutes = 5 x 60 = 300 seconds

(a) Since the resistors are connected in parallel, the voltage across each resistor is the same as the battery voltage, $\text V$ = 24 V.

On using Ohm’s Law $\left (\text I = \dfrac{\text V}{\text R}\right)$,

Then, current through R₁ is given by,

$\text I$1 = \dfrac{\text V}{\text R1} \\[1em] = \dfrac{24}{6} \\[1em] \Rightarrow \text I_1 = 4\ \text A

And, current through R₂ is given by,

$\text I$2 = \dfrac{\text V}{\text R2} \\[1em] = \dfrac{24}{12} \\[1em] \Rightarrow \text I_2 = 2\ \text A

On using Joule’s Law of Heating ($\text H = \text I^2 \text {Rt}$),

Now, heat generated in R₁ is given by,

$\text H$1 = \text I1^2 \times \text R_1 \times \text t \\[1em] = 4^2 \times 6 \times 300 \\[1em] = 16 \times 6 \times 300 \\[1em] = 16 \times 1800 \\[1em] = 28800\ \text J

Similarly,

Heat generated in R₂ is given by,

$\text H$2 = \text I2^2 \times \text R_2 \times \text t \\[1em] = 2^2 \times 12 \times 300 \\[1em] = 4 \times 12 \times 300 \\[1em] = 4 \times 3600 \\[1em] = 14400\ \text J

Total heat generated = $\text H$1 + \text H2 = 28800 + 14400 = 43200 J

So, the total heat generated in both resistor is 43200 J.

(b) Given,

Power rating of each resistor = 100 W

The power dissipated by each resistor can be calculated using $\text P = \text V \times \text I$

Then,

Power dissipated by R₁ is given by,

$\text P$1 = \text V \times \text I1 \\[1em] = 24 \times 4 \\[1em] = 96\ \text W

Similarly,

Power dissipated by R₁ is given by,

$\text P$2 = \text V \times \text I2 \\[1em] = 24 \times 2 \\[1em] = 48\ \text W

As, R₁ is operating at 96 W and R₂ at 48 W, which is within the 100 W limit.

Hence, it is safe to use both R₁ and R₂ in the circuit.