Analyzing Heaters Connected in Series

This question involves analyzing the power dissipation (heat produced) in two different heaters when they are connected in series across a DC voltage supply. The key principle here is understanding how resistance affects current and power in a series circuit.

Understanding the Heater Specifications

We are given the power rating and operating voltage for each heater when connected individually to that voltage:

• Heater 1: 200V, 300W
• Heater 2: 200V, 600W

These ratings allow us to determine the resistance of each heater using the formula relating power (P), voltage (V), and resistance (R):

\(P = \frac{V^2}{R}\)

Rearranging the formula to find resistance:

\(R = \frac{V^2}{P}\)

Calculating the Resistance of Each Heater

Using the formula, we calculate the resistance for Heater 1 (300W) and Heater 2 (600W):

• Resistance of Heater 1 (\(R_1\)):
• \(R_1 = \frac{(200V)^2}{300W} = \frac{40000}{300} \Omega = \frac{400}{3} \Omega\)
• Resistance of Heater 2 (\(R_2\)):
• \(R_2 = \frac{(200V)^2}{600W} = \frac{40000}{600} \Omega = \frac{400}{6} \Omega = \frac{200}{3} \Omega\)

We can see that \(R_1 = \frac{400}{3} \Omega\) and \(R_2 = \frac{200}{3} \Omega\). Therefore, the 300W heater has a higher resistance than the 600W heater.

Heaters Connected in Series

When these two heaters are connected in series to a 200V DC supply, the total resistance of the circuit is the sum of their individual resistances:

\(R_{total} = R_1 + R_2\)

\(R_{total} = \frac{400}{3} \Omega + \frac{200}{3} \Omega = \frac{600}{3} \Omega = 200 \Omega\)

Now, we can calculate the total current flowing through the series circuit using Ohm's Law (\(V = I \times R\)), where \(V\) is the total supply voltage and \(R_{total}\) is the total resistance:

\(I = \frac{V_{supply}}{R_{total}}\)

\(I = \frac{200V}{200 \Omega} = 1A\)

In a series connection, the same current flows through both heaters (Heater 1 and Heater 2).

Calculating Heat Produced in Series Connection

The heat produced by a component in a circuit is equivalent to the power dissipated by it. In a series circuit, where the current \(I\) is the same through both resistors, the power dissipated (P) in each resistor is given by the formula:

\(P = I^2 \times R\)

Let's calculate the power dissipated (heat produced) by each heater when connected in series:

• Heat produced in Heater 1 (\(P_{1,series}\)):
• \(P_{1,series} = I^2 \times R_1 = (1A)^2 \times \frac{400}{3} \Omega = 1 \times \frac{400}{3} W = \frac{400}{3} W\)
• Heat produced in Heater 2 (\(P_{2,series}\)):
• \(P_{2,series} = I^2 \times R_2 = (1A)^2 \times \frac{200}{3} \Omega = 1 \times \frac{200}{3} W = \frac{200}{3} W\)

Comparing Heat Produced

Now we compare the heat produced by the two heaters in the series connection:

\(P_{1,series} = \frac{400}{3} W \)

\(P_{2,series} = \frac{200}{3} W \)

Clearly, \(\frac{400}{3} W > \frac{200}{3} W\). The heater with the higher resistance (\(R_1\), which was originally the 300W heater) produces more heat when connected in series.

This demonstrates that in a series circuit, components with higher resistance dissipate more power (produce more heat) for the same current flowing through them.

Summary Table

The 300W heater (Heater 1) with the higher resistance produces more heat when the heaters are connected in series.