Two heaters A and B have power rating of 1 kW and 2 kW, respectively. Those two are first connected

Two heaters A and B have power rating of 1 kW and 2 kW, respectively. Those two are first connected in series and then in parallel to a fixed power source. The ratio of power outputs for these two cases is :
(1) 1:1
(2) 2:9
(3) 1:2
(4) 2:3

Heaters are electrical devices designed to convert electrical energy into heat. They are often rated by their power output, measured in watts (W) or kilowatts (kW). Understanding how these heaters behave in different electrical configurations is crucial for efficient energy management and practical applications.

#### Series Connection

When heaters are connected in series, they share the same electrical current. The total resistance of the series connection is the sum of the individual resistances. In this configuration, the voltage across each heater is different, depending on its resistance. The power output of each heater in series depends on its resistance and the shared current.

**Resistance and Power**: The heater with higher resistance will have a larger voltage drop across it, leading to a different power output compared to a heater with lower resistance.

#### Parallel Connection

When heaters are connected in parallel, each heater experiences the same voltage across it, but the current through each heater can differ. The total power supplied to the system is the sum of the powers consumed by each heater.

**Power Calculation**: The power output of each heater in parallel depends directly on its rated power and the constant voltage supplied. In parallel configurations, heaters with higher power ratings will consume more power.

#### Comparison of Power Outputs

Comparing heaters in series versus parallel reveals significant differences in their power distribution:

**In Series**: The power output varies with the resistance of each heater. Heaters with different resistances will have different power outputs, and the total power is influenced by the combined resistance.

**In Parallel**: Each heater's power output is directly proportional to its rating since they are all subjected to the same voltage. Thus, the power output for each heater is straightforward and additive.

Understanding these configurations helps in designing circuits that meet specific heating requirements while ensuring efficient use of electrical energy.
Let's analyze the power output of heaters A and B when connected in series and in parallel to a fixed power source.

Case 1: Heaters in Series

When the heaters are connected in series, the same current flows through both heaters. Let's denote the fixed power source voltage as \( V \).

The resistance of heater A (1 kW) is \( R_A \). Since \( P_A = \frac{V^2}{R_A} \), we have \( R_A = \frac{V^2}{P_A} = \frac{V^2}{1000} \).
The resistance of heater B (2 kW) is \( R_B \). Similarly, \( R_B = \frac{V^2}{P_B} = \frac{V^2}{2000} \).

The total resistance \( R_{total} \) in series is:
\[ R_{total} = R_A + R_B = \frac{V^2}{1000} + \frac{V^2}{2000} = \frac{2V^2 + V^2}{2000} = \frac{3V^2}{2000} \]

The current \( I \) through the series circuit is:
\[ I = \frac{V}{R_{total}} = \frac{V}{\frac{3V^2}{2000}} = \frac{2000}{3V} \]

The power output of heater A in series is:
\[ P_A^{series} = I^2 \times R_A = \left(\frac{2000}{3V}\right)^2 \times \frac{V^2}{1000} = \frac{4000000}{9} \times \frac{1}{1000} = \frac{4000}{9} \text{ W} \]

The power output of heater B in series is:
\[ P_B^{series} = I^2 \times R_B = \left(\frac{2000}{3V}\right)^2 \times \frac{V^2}{2000} = \frac{4000000}{9} \times \frac{1}{2000} = \frac{2000}{9} \text{ W} \]

Case 2: Heaters in Parallel

When the heaters are connected in parallel, they have the same voltage \( V \) across them.

The power output of heater A in parallel is \( P_A^{parallel} = P_A = 1000 \text{ W} \)
The power output of heater B in parallel is \( P_B^{parallel} = P_B = 2000 \text{ W} \)

Ratio of Power Outputs

Now we find the ratio of the power outputs for the two cases:


Thus, the ratio of power outputs for the two cases is:
Overview of Heater Power Analysis in Series and Parallel

#### Power Ratings and Electrical Configurations

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