Three resistances of 6Ω each are connected as shown below. What will be the equivalent resistance between points P and Q? Which of the following figures correctly depicts the relat... Three resistances of 6Ω each are connected as shown below. What will be the equivalent resistance between points P and Q? Which of the following figures correctly depicts the relationship between voltage (V) and current (I) as per Ohm's law?
Understand the Problem
The question is asking for the calculation of the equivalent resistance between two points in a circuit with three resistances of equal value. It also asks to identify the correct graphical representation of the relationship between voltage and current as described by Ohm's law.
Answer
The equivalent resistance is $4 \, \Omega$.
Answer for screen readers
The equivalent resistance between points P and Q is $4 , \Omega$.
Steps to Solve
- Identify the configuration of resistors
The three resistors of $6 , \Omega$ each are connected in a combination of series and parallel. The two outer resistors are in series with the parallel combination of the third resistor.
- Calculate the equivalent resistance of parallel resistors
The two outer resistors are in series with one inner resistor. The equivalent resistance of the two outer resistors (each $6 , \Omega$) in series is calculated as: $$ R_{series} = R_1 + R_2 = 6 + 6 = 12 , \Omega $$
- Combine the series result with the parallel resistor
Now we find the equivalent resistance between points P and Q as follows. Let $R_{parallel}$ be the equivalent resistance of the parallel combination, which is: $$ R_{total} = R_{series} + R_{parallel} = 12 , \Omega + 6 , \Omega = 12 , \Omega $$
- Finalize the equivalent resistance
Since the inner resistor is in parallel with the combined resistance of the two resistors, the new equivalent resistance between points P and Q is: $$ R_{eq} = \left( \frac{1}{R_{series}} + \frac{1}{6} \right)^{-1} = \left( \frac{1}{12} + \frac{1}{6} \right)^{-1} $$
Calculating this gives: $$ \frac{1}{R_{eq}} = \frac{1}{12} + \frac{2}{12} = \frac{3}{12} \Rightarrow R_{eq} = \frac{12}{3} = 4 , \Omega $$
- Identify the correct figure for Ohm’s law
According to Ohm's law, the relationship between voltage ($V$) and current ($I$) is linear. Therefore, the correct graphical representation is a straight line passing through the origin which indicates a positive correlation.
The equivalent resistance between points P and Q is $4 , \Omega$.
More Information
The equivalent resistance in a circuit can change based on how resistors are arranged (in series or parallel). Ohm's law states $V = IR$, showing the direct proportionality between voltage and current, which depicts a linear relationship in graphs.
Tips
- Confusing series and parallel combinations. Always double-check the arrangement of resistors.
- Misapplying the formula for combining resistors, particularly in parallel and series configurations. Review the formulas carefully before calculations.
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