To solve the problem of finding the resistances of the three parallel-connected resistors, we can start by analyzing the information given. We know the total power dissipated, the total current drawn, and the current flowing through the first resistor. Let's break this down step by step.
Understanding the Circuit Configuration
In a parallel circuit, the voltage across each resistor is the same, and the total current is the sum of the currents through each resistor. Given that the total current is 6 A and the current through the first resistor (R1) is 3 A, we can determine the current flowing through the second (R2) and third (R3) resistors.
Calculating the Current Through R2 and R3
Since the total current is 6 A and R1 carries 3 A, the combined current through R2 and R3 is:
- Current through R2 + Current through R3 = Total Current - Current through R1
- Let I2 = I3 (since R2 and R3 have equal resistance)
- 2I2 = 6 A - 3 A = 3 A
- I2 = 1.5 A
Using Power to Find Resistance
The power dissipated by each resistor can be calculated using the formula:
P = I²R
We know the total power dissipated is 72 W, which can also be expressed as:
Calculating Power for Each Resistor
For R1, the power is:
P1 = I1²R1 = 3²R1 = 9R1
For R2 and R3, since they have the same current (1.5 A) and resistance (R), the power for each is:
P2 = I2²R2 = (1.5)²R = 2.25R
P3 = I3²R3 = (1.5)²R = 2.25R
Setting Up the Equation
Now we can set up the equation for total power:
9R1 + 2.25R + 2.25R = 72
This simplifies to:
9R1 + 4.5R = 72
Finding the Voltage Across the Resistors
Using Ohm's Law, we can find the voltage (V) across the resistors:
V = I_total × R_total
Since we know the total current (6 A) and we can express the total resistance in terms of R1 and R:
1/R_total = 1/R1 + 1/R + 1/R
Thus, we can rewrite it as:
R_total = R1 || R
Where R1 || R is the equivalent resistance of R1 in parallel with R2 and R3.
Solving for the Resistances
We can express R_total in terms of R1 and R:
1/R_total = 1/R1 + 2/R
Now, we can find the voltage using the total power:
P = V × I_total
So, V = 72 W / 6 A = 12 V.
Now we can use this voltage to find R1:
V = I1 × R1
12 V = 3 A × R1
R1 = 12 V / 3 A = 4 Ω.
Substituting R1 back into the power equation:
9(4) + 4.5R = 72
36 + 4.5R = 72
4.5R = 72 - 36 = 36
R = 36 / 4.5 = 8 Ω.
Final Results
Thus, the resistances of the three resistors are:
- R1 = 4 Ω
- R2 = 8 Ω
- R3 = 8 Ω
Visual Representation
Here’s a simple diagram of the circuit:
This diagram illustrates the three resistors connected in parallel across the voltage source, with the respective currents flowing through each. R1 has a current of 3 A, while R2 and R3 each carry 1.5 A.
In summary, we have successfully calculated the resistances of the three resistors based on the given parameters. If you have any further questions or need clarification on any part of this process, feel free to ask!
