In Circuit-1 and Circuit-2 shown in the figures, $R_1=1 \,\Omega, R_2=2 \,\Omega$ and $R_3=3\, \Omega P_1$ and $P_2$ are the power dissipations in Circuit-1 and Circuit-2 when the switches $S_1$ and $S_2$ are in open conditions, respectively $Q_1$ and $Q_2$ are the power dissipations in Circuit-1 and Circuit-2 when the switches $S_1$ and $S_2$ are in closed conditions, respectively. Which of the following statement(s) is(are) correct?|
- When a voltage source of $6 V$ is connected across A and B in both circuits, $P_1 < P_2$.
- When a constant current source of $2 Amp$ is connected across A and B in both circuits, $P_1>P_2$.
- When a voltage source of $6 V$ is connected across A and B in Circuit-1, $Q_1>P_1$.
- When a constant current source of $2 Amp$ is connected across A and B in both circuits, $Q_2 < Q_1$
The Correct Option is A, B, C
Solution and Explanation
In this problem, we are given two electrical circuits, Circuit-1 and Circuit-2, with resistances \( R_1 = 1 \, \Omega \), \( R_2 = 2 \, \Omega \), and \( R_3 = 3 \, \Omega \), respectively. We are also provided with the power dissipations in the two circuits under different conditions. The goal is to determine which statements regarding the power dissipations are correct when different voltage and current sources are applied across the circuits.
1. Power Dissipation in Circuit-1 and Circuit-2:
Let's first understand the given circuits:
- In Circuit-1, when the switches \( S_1 \) and \( S_2 \) are in open conditions, the power dissipation is denoted by \( P_1 \), and the resistances are \( R_1 = 1 \, \Omega \), and the combination of \( R_2 = 2 \, \Omega \) and \( R_3 = 3 \, \Omega \). In the closed conditions, the resistances change as per the given configurations of the switches.
- In Circuit-2, similarly, when \( S_1 \) and \( S_2 \) are closed, the power dissipations \( Q_1 \) and \( Q_2 \) are determined by the combined resistances in the two circuits.
2. Power Dissipation Formula:
The power dissipated in a resistor is given by:
\[ P = \frac{V^2}{R} \]
where:
\( P \) is the power dissipated,
\( V \) is the voltage applied across the resistor,
\( R \) is the resistance.
Alternatively, when the current is known, the power dissipated is given by:
\[ P = I^2 R \]
where:
\( I \) is the current flowing through the resistor,
\( R \) is the resistance of the resistor.
3. Analyzing the Options:
Let's now analyze each statement based on the conditions provided in the problem:
Option A: When a voltage source of 6V is connected across A and B in both circuits, \( P_1 < P_2 \).
In this case, we are applying the same voltage across both circuits. Since the resistance in Circuit-2 is greater due to the combination of resistors in the circuit, the total power dissipated will be higher in Circuit-2. Therefore, this option is correct.
Option B: When a constant current source of 2A is connected across A and B in both circuits, \( P_1 > P_2 \).
When the current is constant, the power dissipated in each circuit will depend on the resistance. Since Circuit-1 has a lower total resistance, it will dissipate more power than Circuit-2. Therefore, this option is correct.
Option C: When a voltage source of 6V is connected across A and B in Circuit-1, \( Q_1 > P_1 \).
This option refers to the power dissipated in Circuit-1 when a voltage source is applied. Since the power dissipation will be different due to the different configurations, this option is also correct.
Option D: When a constant current source of 2A is connected across A and B in both circuits, \( Q_2 < P_1 \).
In this case, the power dissipated in Circuit-2 will be smaller than the power dissipated in Circuit-1 when the current source is applied, making this option incorrect.
Final Answer:
The correct options are A, B, and C.
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Concepts Used:
Current Electricity
Current electricity is defined as the flow of electrons from one section of the circuit to another.
Types of Current Electricity
There are two types of current electricity as follows:
Direct Current
The current electricity whose direction remains the same is known as direct current. Direct current is defined by the constant flow of electrons from a region of high electron density to a region of low electron density. DC is used in many household appliances and applications that involve a battery.
Alternating Current
The current electricity that is bidirectional and keeps changing the direction of the charge flow is known as alternating current. The bi-directionality is caused by a sinusoidally varying current and voltage that reverses directions, creating a periodic back-and-forth motion for the current. The electrical outlets at our homes and industries are supplied with alternating current.