Math Problem Statement
Solution
This circuit consists of resistors , , and , where and are in parallel, and is in series with this parallel combination. We are tasked with calculating the total current , the branch currents and , and the voltages across and the parallel combination of and (i.e., and ).
Step 1: Equivalent Resistance of and (Parallel combination)
The total resistance of two resistors in parallel, and , is given by: Substituting the given values of and : Thus,
Step 2: Total Resistance of the Circuit
Since is in series with , the total resistance is:
Step 3: Calculate Total Current
Using Ohm's Law, the total current is given by: Substituting and :
Step 4: Voltage Across ()
The voltage across , denoted , is: Substituting and :
Step 5: Voltage Across the Parallel Branch ()
The voltage across the parallel combination of and , denoted , is the same for both resistors because they are in parallel: Substituting and :
Step 6: Currents Through and ( and )
Since and are in parallel, the current through each resistor is given by Ohm's Law.
-
For through :
-
For through :
Thus, the current through both resistors is the same because .
Final Results:
- Total current:
- Current through ():
- Current through ():
- Voltage across ():
- Voltage across the parallel branch ():
Would you like further details or clarifications?
Here are five related questions:
- How does the total resistance change if or is altered?
- What happens to the total current if the total voltage is increased?
- Can you describe how to solve for power dissipation in each resistor?
- How does the parallel resistor configuration affect the total circuit behavior?
- How would the current values change if were decreased?
Tip: Always start by simplifying parallel and series resistances before applying Ohm's Law.
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Math Problem Analysis
Mathematical Concepts
Ohm's Law
Series and Parallel Circuits
Electrical Resistance
Current Division
Formulas
V = IR (Ohm's Law)
R_parallel = (1/R2 + 1/R3)^-1 (Parallel Resistance)
I_total = V_total / R_total (Total Current)
I_1 = V_parallel / R2 (Current in R2)
I_2 = V_parallel / R3 (Current in R3)
Theorems
Ohm's Law
Kirchhoff's Current Law
Suitable Grade Level
Undergraduate - Electrical Engineering or Physics
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