Math Problem Statement
Solution
To solve the circuit problem, we need to determine the voltages across points AC, BC, and the current flowing through . Let's break down the process step by step:
1. Simplifying the Circuit:
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Resistors and are in parallel. Solving for .
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After finding , this resistance is in series with , forming a new total resistance between points and .
Total resistance between and will be:
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Resistors and are in series between points and , so the total resistance between and is:
2. Total Circuit Resistance:
The circuit is essentially a combination of two branches (AC and BC) in parallel. The total resistance of the circuit is calculated as:
3. Finding the Currents:
Using Ohm's law, the total current in the circuit can be found by: Where .
4. Current in Each Branch:
Once the total current is found, we can find the current through the and branches using the current divider rule.
5. Voltage Across AC and BC:
The voltage across and is calculated using Ohm’s law for each branch:
6. Current Through :
The current through is the same as the current in the branch, since and are in series.
Now let me calculate the values.Here are the calculated values for the given circuit:
- The equivalent resistance of and in parallel is .
- The total resistance in the AC branch is .
- The total resistance in the BC branch is .
- The total circuit resistance is .
- The total current in the circuit is approximately .
- The current in the AC branch is .
- The current in the BC branch (and through ) is .
- The voltage across AC is .
- The voltage across BC is also .
This means that the voltage across both branches is equal, which is consistent since they are connected in parallel.
Do you need more detailed explanations, or do you have any other questions?
Related Questions:
- How do you calculate the power dissipated by ?
- What would happen to the total current if was increased?
- How would the circuit change if was removed?
- What is the total power consumption of the circuit?
- How can you verify the current divider rule for this circuit?
Tip:
Always double-check if resistors are in series or parallel before simplifying the circuit to avoid mistakes in calculations.
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Math Problem Analysis
Mathematical Concepts
Ohm's Law
Parallel and Series Circuits
Current Divider Rule
Formulas
V = IR (Ohm's Law)
1 / R_parallel = 1 / R4 + 1 / R5
R_total = R_AC + R_BC
I_total = V_total / R_total
V_AC = I_AC * R_AC
V_BC = I_BC * R_BC
Theorems
Current Divider Rule
Ohm's Law
Suitable Grade Level
Grades 10-12
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