your electronics friend needs a 20-ohm resistor but has only 40-ohm resistors. He tells you that he can combine them to produce a 20-ohm resistor. how?
in parallel.
1/40 + 1/40 = 1/20
To combine two 40-ohm resistors to produce a 20-ohm resistor, your electronics friend can connect them in series. When resistors are connected in series, their resistances add up.
Here's the step-by-step process to achieve a 20-ohm equivalent resistance using two 40-ohm resistors:
1. Connect one end of the first 40-ohm resistor to one end of the second 40-ohm resistor.
2. Connect the unconnected end of the first resistor to the positive terminal of the circuit or power source.
3. Connect the unconnected end of the second resistor to the negative terminal of the circuit or power source.
By connecting the resistors in series, the total resistance is the sum of the individual resistances. In this case, 40 ohms + 40 ohms equals 80 ohms, which is not the desired 20-ohm resistor.
To achieve the desired 20-ohm resistance, your friend can use an additional resistor in parallel with the two 40-ohm resistors. Let's assume this additional resistor has a resistance of R.
Here's how to calculate the value of the additional resistor:
1. Determine the total resistance required (20 ohms).
2. Calculate the total resistance of the two 40-ohm resistors in series (80 ohms).
3. Calculate the resistance needed in parallel to achieve the desired total resistance. The formula for resistors in parallel is:
1/R_parallel = 1/R_1 + 1/R_2
Substitute the known values:
1/R_parallel = 1/80 + 1/R
4. Solve the equation to find the value of R_parallel:
R_parallel = 1 / (1/80 + 1/R)
Simplifying further:
R_parallel = 1 / ((R + 80) / (80R))
R_parallel = 80R / (R + 80)
5. Solve the equation when R_parallel = 20 ohms:
20 = 80R / (R + 80)
6. Cross multiply and solve the resulting equation:
20R + 1600 = 80R
60R = 1600
R = 1600 / 60
R ≈ 26.67 ohms
Therefore, your friend should add a 26.67-ohm resistor in parallel with the two 40-ohm resistors to achieve a total resistance of approximately 20 ohms.
To combine the 40-ohm resistors and create a 20-ohm resistor, your electronics friend can connect two resistors in either series or parallel. Let's explore both options:
1. Series Connection:
When resistors are connected in series, their resistances add up. To achieve a total resistance of 20 ohms using two 40-ohm resistors in series, follow these steps:
- Connect one end of the first 40-ohm resistor to one end of the second 40-ohm resistor.
- Connect the other end of the first resistor to the positive terminal of the circuit, and the other end of the second resistor to the negative terminal of the circuit.
- The combined resistance of the two resistors in series is now 40 + 40 = 80 ohms.
- To reduce it to 20 ohms, your friend can add an additional 40-ohm resistor in series. The total resistance will then be 40 + 40 + 40 = 120 ohms.
2. Parallel Connection:
When resistors are connected in parallel, their reciprocals add up. To achieve a total resistance of 20 ohms using two 40-ohm resistors in parallel, follow these steps:
- Connect one end of the first 40-ohm resistor to one end of the second 40-ohm resistor.
- Connect the other end of both resistors to the positive terminal of the circuit.
- The combined resistance of the two resistors in parallel can be calculated as:
1 / (1/40 + 1/40) = 1 / (0.025 + 0.025) = 1 / 0.05 = 20 ohms.
Therefore, your friend can connect two 40-ohm resistors in parallel to achieve a total resistance of 20 ohms.