Math Problem Statement
Solution
This problem involves applying Kirchhoff's laws to determine the current intensities , , and in the given circuit.
Kirchhoff's Laws:
- Kirchhoff's Current Law (KCL): The sum of currents entering a junction is equal to the sum of currents leaving the junction.
- Kirchhoff's Voltage Law (KVL): The sum of the potential differences (voltage drops and gains) around any closed loop is zero.
Applying Ohm’s Law:
Ohm’s Law: relates the voltage , current , and resistance .
Circuit Analysis
The circuit has two voltage sources:
- 8V on the left, and
- 9V on the bottom.
It has three resistors:
- 2 ohms,
- 3 ohms,
- 4 ohms.
Step 1: Define Loop Currents
Let:
- be the current through the 3 ohm resistor (left side),
- be the current through the 2 ohm resistor (middle branch),
- be the current through the 4 ohm resistor (right side).
Step 2: Kirchhoff’s Voltage Law (KVL)
We apply KVL to each loop:
Loop 1 (Left Loop):
- Contains the 8V generator, the 2 ohm resistor, and the 3 ohm resistor.
- The KVL equation for this loop is:
Loop 2 (Right Loop):
- Contains the 9V generator, the 2 ohm resistor, and the 4 ohm resistor.
- The KVL equation for this loop is:
Loop 3 (Total Loop Using KCL at the junction):
At the middle node, applying Kirchhoff's Current Law:
Step 3: Solve the Equations
Now we solve the system of equations:
Let me calculate the values of , , and .The currents in the circuit are:
- ,
- ,
- .
These are the intensities of the currents in the circuit as per Kirchhoff's laws and Ohm's law.
Do you have any questions or would you like further explanation?
Here are some related questions:
- How do Kirchhoff's laws apply to complex circuits with multiple loops?
- What are some common mistakes when using Kirchhoff's laws in circuit analysis?
- How would the current values change if one of the resistances increased?
- Can you explain how to identify junctions and loops in a circuit diagram?
- How does the direction of current affect the sign in Kirchhoff’s Voltage Law?
Tip: Always check that the sum of voltage drops equals the voltage supplied when applying KVL, and ensure all current entering and leaving a node balance for KCL!
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Math Problem Analysis
Mathematical Concepts
Circuit Analysis
Ohm's Law
Kirchhoff's Current Law (KCL)
Kirchhoff's Voltage Law (KVL)
Formulas
Ohm's Law: U = IR
Kirchhoff's Current Law: Σ I (in) = Σ I (out)
Kirchhoff's Voltage Law: Σ V = 0 in a closed loop
Theorems
Kirchhoff's Current Law
Kirchhoff's Voltage Law
Suitable Grade Level
College Level (Advanced High School Physics)
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