Three resistors of different values can be connected together in different configurations.
If all three resistors are used in a circuit, how many different values of total resistance can be
produced?
I can only find five... but the answer key says eight.
series,
then parallel,
then one in series with two parallel two at a time(I get three combinations here)
so far 5, then
two in series, one in parallel (three ways here).
eight total
To determine the number of different values of total resistance that can be produced when three resistors are connected together, let's consider the possible configurations.
1. Series Connection:
In a series circuit, the resistors are connected one after another, forming a single path for the current to flow through. The total resistance (RT) in a series connection is the sum of the individual resistances (R1, R2, R3).
RT = R1 + R2 + R3
2. Parallel Connection:
In a parallel circuit, the resistors are connected side by side, providing multiple paths for the current to flow through. The total resistance (RT) in a parallel connection is given by the reciprocal of the sum of the reciprocals of the individual resistances (R1, R2, R3).
1/RT = 1/R1 + 1/R2 + 1/R3
Now, let's consider the different values of resistors that can be used.
Assume R1 = 1 ohm, R2 = 2 ohms, and R3 = 3 ohms.
1. Series Connection:
RT = 1 + 2 + 3 = 6 ohms
2. Parallel Connection:
1/RT = 1/1 + 1/2 + 1/3 = 11/6
RT = 6/11 ohms (approximately 0.545 ohms)
By changing the values of resistors, let's explore other possible configurations:
1. R1 = 1 ohm, R2 = 3 ohms, R3 = 2 ohms
Series Connection: RT = 1 + 3 + 2 = 6 ohms
Parallel Connection: RT = 6/11 ohms (approximately 0.545 ohms)
2. R1 = 2 ohms, R2 = 1 ohm, R3 = 3 ohms
Series Connection: RT = 2 + 1 + 3 = 6 ohms
Parallel Connection: RT = 2/3 ohms (approximately 0.67 ohms)
3. R1 = 2 ohms, R2 = 3 ohms, R3 = 1 ohm
Series Connection: RT = 2 + 3 + 1 = 6 ohms
Parallel Connection: RT = 6/11 ohms (approximately 0.545 ohms)
4. R1 = 3 ohms, R2 = 1 ohm, R3 = 2 ohms
Series Connection: RT = 3 + 1 + 2 = 6 ohms
Parallel Connection: RT = 6/11 ohms (approximately 0.545 ohms)
5. R1 = 3 ohms, R2 = 2 ohms, R3 = 1 ohm
Series Connection: RT = 3 + 2 + 1 = 6 ohms
Parallel Connection: RT = 6/11 ohms (approximately 0.545 ohms)
6. R1 = 1 ohm, R2 = 1 ohm, R3 = 4 ohms
Series Connection: RT = 1 + 1 + 4 = 6 ohms
Parallel Connection: RT = 4/6 ohms (approximately 0.67 ohms)
7. R1 = 1 ohm, R2 = 4 ohms, R3 = 1 ohm
Series Connection: RT = 1 + 4 + 1 = 6 ohms
Parallel Connection: RT = 2/3 ohms (approximately 0.67 ohms)
8. R1 = 4 ohms, R2 = 1 ohm, R3 = 1 ohm
Series Connection: RT = 4 + 1 + 1 = 6 ohms
Parallel Connection: RT = 2/3 ohms (approximately 0.67 ohms)
Hence, by considering different combinations of resistor values, we find that there are eight different values of total resistance that can be produced when all three resistors are used in a circuit.
To understand how to arrive at the answer of eight different values of total resistance, let's discuss the different configurations in which the three resistors can be connected together.
1. Series Connection:
When the resistors are connected in series, their resistances add up. Therefore, the total resistance in this configuration is the sum of the individual resistances. Let's denote the three resistors as R1, R2, and R3. The total resistance in series is R_total = R1 + R2 + R3.
2. Parallel Connection:
When the resistors are connected in parallel, the reciprocal of their resistances adds up. Therefore, the total resistance in this configuration is the reciprocal of the sum of the reciprocals of the individual resistances. In this case, the total resistance is given by the equation: 1/R_total = 1/R1 + 1/R2 + 1/R3.
Now, let's explore the possibilities for different values of total resistance.
CASE 1: All three resistors are different.
In this case, we have three different resistors (R1, R2, R3), and we can arrange them in any order in both series and parallel connections. So, the possible combinations are:
1. Series Connection: R1-R2-R3
2. Series Connection: R1-R3-R2
3. Series Connection: R2-R1-R3
4. Series Connection: R2-R3-R1
5. Series Connection: R3-R1-R2
6. Series Connection: R3-R2-R1
Each of these combinations yields a different value of total resistance.
CASE 2: Two resistors are the same, and one resistor is different.
In this case, we have one resistor that is different from the other two. Let's assume R1 = R2 and R3 is different. Now, we can place R3 in series with the identical resistors OR in parallel with the identical resistors. So, the possible combinations are:
1. Series Connection: R1-R2-R3 (R1 and R2 are the same)
2. Parallel Connection: R1||R2-R3 (R1 and R2 are the same)
Each of these combinations yields a different value of total resistance.
CASE 3: All three resistors are the same.
In this case, all three resistors are identical, so there is only one possible combination:
1. Series Connection: R1-R1-R1 (R1 is the same for all three resistors)
This combination yields a different value of total resistance.
Therefore, by considering all the possibilities from the three cases, we obtained eight different values of total resistance. If you have found only five values of total resistance, it's possible that you missed considering all the permutations and combinations for the different resistor configurations.