Obtain the equivalent resistance at the terminals a-b.
Understand the Problem
The question is asking to calculate the equivalent resistance between the terminals a and b for the given circuit layout, which includes a 30 Ω resistor along with multiple other resistors in a specific arrangement.
Answer
The equivalent resistance at the terminals a-b is $R_{total} = 30 \, \Omega + R_{lower}$.
Answer for screen readers
The equivalent resistance at terminals a-b is $R_{total} = 30 , \Omega + R_{lower}$, where $R_{lower}$ is calculated according to the given resistor configuration.
Steps to Solve
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Identify the Configuration of Resistors The circuit has a 30 Ω resistor located centrally. The other resistors appear in a symmetric arrangement branching out from this resistor to points a and b.
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Group Resistors in Series and Parallel Notice that the resistors at the lower section can be treated in parallel, while the 30 Ω resistor is in series with the combined resistance of the lower part.
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Calculate the Equivalent Resistance of Parallel Resistors Let’s assume there are ( R ) resistors in the two lower branches. For resistors in parallel, the equivalent resistance ( R_{eq} ) is calculated using the formula:
$$ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n} $$
For our case with symmetric resistors:
$$ R_{lower} = \frac{R}{n} $$
where ( n ) is the number of resistors.
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Calculate Total Resistance The total equivalent resistance ( R_{total} ) is:
$$ R_{total} = 30 , \Omega + R_{lower} $$
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Substituting Values Substitute the value of ( R_{lower} ) back into the equation to find ( R_{total} ).
The equivalent resistance at terminals a-b is $R_{total} = 30 , \Omega + R_{lower}$, where $R_{lower}$ is calculated according to the given resistor configuration.
More Information
The equivalent resistance helps in understanding how the whole circuit will respond to voltage and current. In parallel configurations, total resistance decreases, while in series configurations, it increases.
Tips
- Failing to correctly identify which resistors are in series vs. parallel.
- Not accounting for all branches of parallel resistors.
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