Question

Eight different values of resistance can be obtained by connecting together three resistors 6.50 Ω, 7.60 Ω, and 1.70 Ω in all possible ways. What are the values in the following situations? All the resistors are connected in series. Ω All the resistors are connected in parallel. Ω The 6.50 Ω and 7.60 Ω resistors are connected in parallel, and the 1.70-Ω resistor is connected in series with the parallel combination. Ω The 6.50 Ω and 1.70 Ω resistors are connected in parallel, and the 7.60-Ω resistor is connected in series with the parallel combination. Ω The 7.60 Ω and 1.70 Ω resistors are connected in parallel, and the 6.50-Ω resistor is connected in series with the parallel combination. Ω The 6.50 Ω and 7.60 Ω resistors are connected in series, and the 1.70-Ω resistor is connected in parallel with the series combination. Ω The 6.50 Ω and 1.70 Ω resistors are connected in series, and the 7.60-Ω resistor is connected in parallel with the series combination. Ω The 7.60 Ω and 1.70 Ω resistors are connected in series, and the 6.50-Ω resistor is connected in parallel with the series combination. Ω Additional Materials

Answer 1

Before starting with the problem, let's remind that:

- Total resistance of a combination of resistors in series is:

$R_T=R_1+R_2+....$

- Total resistance of a combination of resistors in parallel is given by:

$\frac{1}{R_T}=\frac{1}{R_1}+\frac{1}{R_2}+...$

Now let's apply these equations to solve the different parts of the problem:

1. All the resistors in series: $15.8 \Omega$

In this case, the total resistance is

$R_T=6.50\Omega +7.60\Omega+1.70\Omega=15.8 \Omega$

2. All the resistors are connected in parallel: $1.14 \Omega$

In this case, the total resistance is

$\frac{1}{R_T}=\frac{1}{6.50 \Omega}+\frac{1}{7.60 \Omega}+\frac{1}{1.70 \Omega}=0.874\Omega^{-1}\\R_T=\frac{1}{0.874 \Omega^{-1}}=1.14 \Omega$

3. The 6.50 Ω and 7.60 Ω resistors are connected in parallel, and the 1.70-Ω resistor is connected in series with the parallel combination: $5.20 \Omega$

The total resistance of the two resistors connected in parallel is

$\frac{1}{R_T}=\frac{1}{6.50 \Omega}+\frac{1}{7.60\Omega}=0.285\Omega^{-1}\\R_T=\frac{1}{0.285 \Omega^{-1}}=3.50 \Omega$

And the combination of these with the other resistor of 1.70-Ω in series gives a total resistance of

$R_T=3.50 \Omega+1.70 \Omega=5.20 \Omega$

4. The 6.50 Ω and 1.70 Ω resistors are connected in parallel, and the 7.60-Ω resistor is connected in series with the parallel combination: $8.95 \Omega$

The total resistance of the two resistors connected in parallel is

$\frac{1}{R_T}=\frac{1}{6.50 \Omega}+\frac{1}{1.70\Omega}=0.742\Omega^{-1}\\R_T=\frac{1}{0.742 \Omega^{-1}}=1.35 \Omega$

And the combination of these with the other resistor of 7.60-Ω in series gives a total resistance of

$R_T=1.35 \Omega+7.60 \Omega=8.95 \Omega$

5. The 7.60 Ω and 1.70 Ω resistors are connected in parallel, and the 6.50-Ω resistor is connected in series with the parallel combination: $7.89\Omega$

The total resistance of the two resistors connected in parallel is

$\frac{1}{R_T}=\frac{1}{7.60 \Omega}+\frac{1}{1.70\Omega}=0.720\Omega^{-1}\\R_T=\frac{1}{0.720 \Omega^{-1}}=1.39 \Omega$

And the combination of these with the other resistor of 6.50-Ω in series gives a total resistance of

$R_T=1.39 \Omega+6.50 \Omega=7.89 \Omega$

6. The 6.50 Ω and 7.60 Ω resistors are connected in series, and the 1.70-Ω resistor is connected in parallel with the series combination: $1.52 \Omega$

In this case, the total resistance of the two resistors in series is

$R_T=6.50\Omega + 7.60\Omega=14.10 \Omega$

And the combination of these with the other resistor of 1.70-Ω in parallel gives a total resistance of

$\frac{1}{R_T}=\frac{1}{14.10 \Omega}+\frac{1}{1.70\Omega}=0.659\Omega^{-1}\\R_T=\frac{1}{0.720 \Omega^{-1}}=1.52 \Omega$

7. The 6.50 Ω and 1.70 Ω resistors are connected in series, and the 7.60-Ω resistor is connected in parallel with the series combination: $3.94 \Omega$

In this case, the total resistance of the two resistors in series is

$R_T=6.50\Omega + 1.70\Omega=8.20 \Omega$

And the combination of these with the other resistor of 7.60-Ω in parallel gives a total resistance of

$\frac{1}{R_T}=\frac{1}{8.20 \Omega}+\frac{1}{7.6\Omega}=0.254\Omega^{-1}\\R_T=\frac{1}{0.720 \Omega^{-1}}=3.94 \Omega$

8. The 7.60 Ω and 1.70 Ω resistors are connected in series, and the 6.50-Ω resistor is connected in parallel with the series combination: $3.83\Omega$

In this case, the total resistance of the two resistors in series is

$R_T=7.60\Omega + 1.70\Omega=9.30 \Omega$

And the combination of these with the other resistor of 6.50-Ω in parallel gives a total resistance of

$\frac{1}{R_T}=\frac{1}{9.30 \Omega}+\frac{1}{6.50\Omega}=0.261\Omega^{-1}\\R_T=\frac{1}{0.720 \Omega^{-1}}=3.83 \Omega$