Math Problem Statement
Solution
To determine the current in each branch using nodal analysis for the given circuit, let's first outline the steps:
Step 1: Identify Nodes
Let’s assign nodes for clarity:
- Node 1: The top point between the 3Ω, 10Ω, and 5Ω resistors.
- Ground: The bottom of the circuit (reference node).
Step 2: Apply KCL (Kirchhoff's Current Law) at each node
At each node, the sum of the currents leaving the node is zero.
For Node 1:
Let the voltage at Node 1 be with respect to the reference node (ground).
Using Ohm’s law for each branch, we express the currents in terms of :
-
Current through the 3Ω resistor:
-
Current through the 10Ω resistor (connected between Node 1 and the 5A current source): Since there is a 5A current source on the left, the current through the 10Ω resistor must balance with that current source:
-
Current through the 5Ω resistor (connected between Node 1 and the 10A current source): This is the current flowing through the 5Ω resistor due to the 10A current source on the right:
Now, applying KCL at Node 1:
Step 3: Simplify the Equation
To simplify, let’s first multiply through by the least common denominator, which is 30:
Now combine like terms:
Solve for :
Step 4: Calculate the Currents in Each Branch
Now that we have , we can calculate the currents through each resistor:
-
Current through the 3Ω resistor:
-
Current through the 10Ω resistor:
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Current through the 5Ω resistor:
Final Answer:
- Current through 3Ω: (negative indicates direction opposite to assumed).
- Current through 10Ω: .
- Current through 5Ω: .
Let me know if you'd like a detailed explanation for any of these steps!
Follow-up Questions:
- How is the polarity of voltages determined in nodal analysis?
- What happens if one of the resistances is replaced with a capacitor?
- How would the current values change if the 5A current source was changed?
- Can you explain the impact of having more nodes in this circuit?
- How would mesh analysis differ in solving this circuit?
Tip:
In nodal analysis, always make sure to correctly set up the ground (reference) and ensure consistency in assuming the direction of currents.
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Math Problem Analysis
Mathematical Concepts
Circuit Analysis
Nodal Analysis
Kirchhoff's Current Law (KCL)
Ohm's Law
Formulas
Ohm's Law: V = IR
Kirchhoff's Current Law: Sum of currents at a node = 0
Theorems
Kirchhoff's Current Law (KCL)
Suitable Grade Level
Grades 11-12
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