Find the total resistance of the network of the figure, where RA = 3 Ω, RB = 3 Ω, and RC = 6 Ω. Find the resistance R1.
Understand the Problem
The question is asking to calculate the total resistance (RT) of a network of resistors given specific values for resistors RA, RB, and RC. The overall approach involves using formulas for resistors in series and parallel to derive the total resistance from the given values.
Answer
The total resistance \( R_T \) is \( 5 \, \Omega \).
Answer for screen readers
The total resistance ( R_T ) of the network is ( 5 , \Omega ).
Steps to Solve
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Identify resistor combinations
The resistors in the circuit include ( R_A = 3 , \Omega ), ( R_B = 3 , \Omega ), and ( R_C = 6 , \Omega ). We need to analyze how these resistors are connected (series or parallel). -
Calculate equivalent resistances for R_B and R_C
Resistors ( R_B ) (3 Ω) and ( R_C ) (6 Ω) are in parallel. The formula for equivalent resistance ( R_{eq} ) for resistors in parallel is given by:
$$ \frac{1}{R_{eq}} = \frac{1}{R_B} + \frac{1}{R_C} $$
Substituting the values:
$$ \frac{1}{R_{eq}} = \frac{1}{3} + \frac{1}{6} $$
Finding a common denominator (6):
$$ \frac{1}{R_{eq}} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} $$
Thus,
$$ R_{eq} = 2 , \Omega $$ -
Combine R_A and R_eq in series
Now, ( R_A ) (3 Ω) is in series with ( R_{eq} ) (2 Ω):
The total resistance ( R_T ) is:
$$ R_T = R_A + R_{eq} = 3 + 2 = 5 , \Omega $$ -
Final Total Resistance
The total resistance of the network is ( R_T = 5 , \Omega ).
The total resistance ( R_T ) of the network is ( 5 , \Omega ).
More Information
In this circuit, we combined series and parallel resistors to find the total resistance. It's common to first simplify parallel resistors before adding them to those in series.
Tips
- Forgetting to convert the fractions when adding parallel resistances can lead to incorrect ( R_{eq} ) values. Always ensure proper addition and find a common denominator.
- Not distinguishing between series and parallel combinations can result in miscalculating the total resistance.
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