You are given five identical resistors each of resistance r How can you arrange all these resistors (in series in parallel or both) in order to have an equivalent resistance of 7r/2, 7r/3, or 2r?
2 in series. 3 in parallel?
To find different arrangements of resistors that result in the given equivalent resistances, we need to understand the concepts of resistors in series and parallel.
In a series circuit, resistors are connected end-to-end, like a single line. The total resistance (Rs) of resistors in series is the sum of their individual resistances.
In a parallel circuit, resistors are connected to each other in a branching path, like the rungs of a ladder. The total resistance (Rp) of resistors in parallel can be calculated using the formula:
1/Rp = 1/R1 + 1/R2 + 1/R3 + ...
Now let's explore the different arrangements:
1. To achieve an equivalent resistance of 7r/2:
- Series Circuit:
Combine two resistors in series:
Rs = r + r = 2r
Then connect those two resistors in parallel with a single resistor:
Rp = 1/(1/Rs + 1/r) = 1/(1/2r + 1/r) = 1/(3/2r) = 2r/3
2. To achieve an equivalent resistance of 7r/3:
- Series Circuit:
Combine three resistors in series:
Rs = r + r + r = 3r
Then connect those three resistors in parallel with two resistors:
Rp = 1/(1/Rs + 1/(2r)) = 1/(1/3r + 1/(2r)) = 1/(5/6r) = 6r/5 = 7r/5
3. To achieve an equivalent resistance of 2r:
- Parallel Circuit:
Connect all five resistors in parallel:
Rp = 1/(1/r + 1/r + 1/r + 1/r + 1/r) = 1/(5/r) = r/5
Remember, these are just a few possible arrangements. You can experiment and come up with other combinations by understanding the concepts of series and parallel arrangements of resistors.