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
r/2 + 3r = r/2 + 6r/2 = 7r/2 = 2 in parallel and 3 in series.
r/3 + 2r = r/3 + 6r/3 = 7r/3 = 3 in parallel and 2 in series.
r/2 + r/2 + r = 2r = 2 in parallel + 2 in parallel + 1 in series.
To find the required arrangements of five identical resistors to achieve an equivalent resistance of 7r/2, 7r/3, or 2r, we can use a combination of series and parallel connections. Let's go through each case step-by-step:
1. Equivalent Resistance of 7r/2:
To achieve an equivalent resistance of 7r/2, we can connect three resistors in parallel and the remaining two resistors in series. Here's the step-by-step process:
Step 1: Take three resistors and connect them in parallel.
- Equivalent resistance of three resistors connected in parallel (R_parallel) = r/3
Step 2: Take the remaining two resistors and connect them in series.
- Equivalent resistance of two resistors connected in series (R_series) = 2r
Step 3: Connect R_parallel and R_series in series.
- Equivalent resistance (R_eq) = R_parallel + R_series = r/3 + 2r = 7r/3
2. Equivalent Resistance of 7r/3:
To achieve an equivalent resistance of 7r/3, we can connect four resistors in parallel and the remaining resistor separately in parallel. Here's the step-by-step process:
Step 1: Take four resistors and connect them in parallel.
- Equivalent resistance of four resistors connected in parallel (R_parallel) = r/4
Step 2: Take the remaining resistor (not included in R_parallel) and connect it separately in parallel.
- Equivalent resistance of one resistor connected in parallel (R_single_parallel) = r
Step 3: Connect R_parallel and R_single_parallel in series.
- Equivalent resistance (R_eq) = R_parallel + R_single_parallel = r/4 + r = 5r/4 = 7r/3
3. Equivalent Resistance of 2r:
To achieve an equivalent resistance of 2r, we can connect all five resistors in series. Here's the step-by-step process:
Step 1: Take all five resistors and connect them in series.
- Equivalent resistance (R_eq) = 5r
By following these steps, you can arrange the five identical resistors to achieve the desired equivalent resistance.
To find the arrangement of resistors that yields an equivalent resistance of 7r/2, 7r/3, or 2r, we need to consider the combinations of resistors in series and parallel.
1. Equivalent Resistance of 7r/2:
- Series Combination: Connect all five resistors in series. The equivalent resistance will be the sum of the individual resistances, which is 5r.
- Parallel Combination: Connect all five resistors in parallel. The equivalent resistance can be found using the formula: 1/Req = 1/R1 + 1/R2 + 1/R3 + 1/R4 + 1/R5. Since all resistors are identical, we can simplify this to 1/Req = 5/r. By solving this equation, we find that Req = r/5.
To get an equivalent resistance of 7r/2, you can combine the series and parallel combination as follows:
- Connect three resistors in series, which gives a resistance of 3r.
- Connect the remaining two resistors in parallel to each other and the series combination.
- The equivalent resistance of this arrangement will be 3r + (r/5 || r/5), where "||" represents parallel combination. Solving this equation, we get 7r/2.
2. Equivalent Resistance of 7r/3:
- Series Combination: Connect three resistors in series. The equivalent resistance will be 3r.
- Parallel Combination: Connect two resistors in parallel. The equivalent resistance can be found as 1/Req = 1/R1 + 1/R2. Since all resistors are identical, this simplifies to 1/Req = 2/r. By solving this equation, we get Req = r/2.
To obtain an equivalent resistance of 7r/3, you can combine the series and parallel combination as follows:
- Connect two resistors in series, giving a resistance of 2r.
- Connect the remaining three resistors in parallel to each other and the series combination.
- The equivalent resistance of this arrangement will be 2r + (r/2 || r/2 || r/2), where "||" represents parallel combination. Solving this equation, we get 7r/3.
3. Equivalent Resistance of 2r:
- Both Series and Parallel Combination: Connect all five resistors in series and in parallel with each other.
- Connect all five resistors in series, giving an equivalent resistance of 5r.
- Connect all five resistors in parallel, using the formula mentioned above (1/Req = 5/r), we find that Req = r/5.
- The equivalent resistance of this arrangement will be 5r || (r/5), where "||" represents parallel combination. Solving this equation, we get 2r.
In summary, to obtain an equivalent resistance of 7r/2, you can combine three resistors in series and connect the remaining two resistors in parallel to each other and the series combination. For an equivalent resistance of 7r/3, combine two resistors in series and connect the remaining three resistors in parallel to each other and the series combination. Finally, for an equivalent resistance of 2r, connect all five resistors both in series and in parallel.