Question

In: Physics

For the combination of resistors shown, find the equivalent resistance between points A and B.

Part D
For the combination of resistors shown, find the equivalent resistance between points A and B.

(Figure 6)
Express your answer in Ohms.
Req =
 
  ?  

Solutions

Expert Solution

Concepts and reason

The concepts used to solve this problem are effective resistance of resistors connected in series and parallel. First, calculate the effective resistance of the resistors in top and bottom branch connected in parallel. Now, the effective resistances of the parallel resistors are in series with the rest of the resistors in their respective branches. Then, calculate the effective resistance of the series combinations of resistors in top and bottom branch. Now, the equivalent resistance of the top and bottom branch will be parallel to each other. Finally, use the formula to calculate the resistance for the resistors connected in parallel to calculate the equivalent resistance of the resistors in top and bottom branch between point \(A\) and \(B\).

Fundamentals

The expression for the equivalent resistance of the resistor connected in series is, \(R_{\mathrm{eq}}=R_{1}+R_{2}\)

Here, \(R_{\mathrm{eq}}\) is the equivalent resistance of the resistors connected in series, \(R_{1}, R_{2}\) are the resistors connected in series.

The expression for the equivalent resistance of the resistor connected in parallel is, \(\frac{1}{R^{\prime} \mathrm{eq}}=\frac{1}{R 3}+\frac{1}{R 4}+\frac{1}{R 5}\)

Here, \(R_{\mathrm{eq}}^{\prime}\) is the equivalent resistance of the resistors connected in parallel, \(R_{3}, R_{4}, R_{5}\) are the resistors connected in parallel.

 

(D) The given combination of resistors can be divided into top and bottom branch of resistors. The top branch resistors are shown in the figure below:

The expression to calculate the equivalent resistance of the resistor connected in parallel to the top branch is, \(\frac{1}{R_{\mathrm{eqt}}^{\prime}}=\frac{1}{R_{3}}+\frac{1}{R_{4}}+\frac{1}{R_{5}}\)

Here, \(R_{\mathrm{eqt}}^{\prime}\) is the equivalent resistance for three resistors connected in parallel to the top branch in the circuit and \(R_{3}, R_{4}, R_{5}\) are the resistors connected in parallel combination. Substitute \(4 \Omega\) for \(R_{3}, 6 \Omega\) for \(R_{4},\) and \(12 \Omega\) for \(R_{5}\) to find \(R_{\mathrm{eqt}}^{\prime}\)

$$ \begin{array}{c} \frac{1}{R_{\text {eqt }}^{\prime}}=\frac{1}{4 \Omega}+\frac{1}{6 \Omega}+\frac{1}{12 \Omega} \\ =\frac{1}{2 \Omega} \\ R_{\text {eqt }}^{\prime}=2 \Omega \end{array} $$

The expression for the equivalent resistance of the resistor connected in series at the top branch is, \(R_{\mathrm{top}}=R_{1}+R_{2}+R_{\mathrm{eqt}}^{\prime}\)

Substitute \(1 \Omega\) for \(R_{1}, 3 \Omega\) for \(R_{2}, 2 \Omega\) for \(R_{\mathrm{eqt}}^{\prime}\)

$$ \begin{array}{c} R_{\mathrm{top}}=1 \Omega+3 \Omega+2 \Omega \\ =6 \Omega \end{array} $$

The effective resistance of the parallel combination of resistors at the top branch is calculated using the expression \(\frac{1}{R^{\prime} \text { eqt }}=\frac{1}{R_{3}}+\frac{1}{R_{4}}+\frac{1}{R_{5}} .\) It is found that the effective resistance of the capacitors connected in parallel is less than the resistance of the individual resistor in that parallel combination. The effective resistance is now in series with the other resistors at the top branch. Finally, the equivalent resistance of the top branch is found using the expression, \(R_{\mathrm{top}}=R_{1}+R_{2}+R_{\mathrm{eqt}}^{\prime}\)

The bottom branch resistors are shown in the figure below:

The expression for the equivalent resistance of the resistor connected in parallel at the bottom branch is, \(\frac{1}{R^{\prime} \text { eqb }}=\frac{1}{R 6}+\frac{1}{R 7}\)

Here, \(R_{\mathrm{eqb}}^{\prime}\) is the equivalent resistance for two resistors at the bottom branch of the circuit and \(R_{6}, R_{7}\) are the resistors connected in parallel combination. Substitute \(5 \Omega\) for \(R_{6}\) and \(20 \Omega\) for \(R_{7}\) to find \(R_{\mathrm{eqb}}^{\prime}\)

\(\frac{1}{R^{\prime} \text { eqb }}=\frac{1}{R_{6}}+\frac{1}{R_{7}}\)

\(\frac{1}{R^{\prime} \text { eqb }}=\frac{1}{5 \Omega}+\frac{1}{20 \Omega}\)

\(=\frac{1}{4 \Omega}\)

\(R_{\mathrm{eqb}}^{\prime}=4 \Omega\)

The expression for the equivalent resistance of the resistor connected in series at the bottom branch is, \(R_{\text {bottom }}=R_{\mathrm{eqb}}^{\prime}+R_{8}\)

Substitute \(4 \Omega\) for \(R_{\mathrm{eqb}}^{\prime}\) and \(2 \Omega\) for \(R_{8}\) in the above expression.

\(R_{\text {bottom }}=4 \Omega+2 \Omega\)

\(=6 \Omega\)

Now, the equivalent resistance between point \(A\) and \(B\) is,

 

The expression to calculate the equivalent resistance for parallel combination of resistors between point \(\mathrm{A}\) and \(\mathrm{B}\) is, \(\frac{1}{R_{E Q}}=\frac{1}{R_{\mathrm{top}}}+\frac{1}{R_{\mathrm{bottom}}}\)

Substitute \(6 \Omega\) for \(R_{\text {top }}\) and \(6 \Omega\) for \(R_{\text {bottom }}\) in the above expression.

$$ \begin{aligned} \frac{1}{R_{E Q}} &=\frac{1}{6 \Omega}+\frac{1}{6 \Omega} \\ =& \frac{(6 \Omega)(6 \Omega)}{(6 \Omega+6 \Omega)} \\ &=3 \Omega \end{aligned} $$

Part D

Thus, the equivalent resistance between points \(\mathrm{A}\) and \(\mathrm{B}\) is \(3 \Omega\).

The effective resistance of the parallel combination of resistors at the bottom of the circuit is calculated and its effective resistance is found. Finally, the equivalent resistance between points \(\mathrm{A}\) and \(\mathrm{B}\) is calculated using the expression \(\frac{1}{R E Q}=\frac{1}{R_{\text {top }}}+\frac{1}{R \text { bottom }}\)

Related Solutions

What is the equivalent resistance between points a and b of the six resistors shown in the Figure ?
What is the equivalent resistance between points a and b of the six resistors shown in the Figure below?  
Three resistors connected in parallel have an equivalent resistance of 1Ω
Three resistors connected in parallel have an equivalent resistance of 1Ω. If two of the resistance values are 2Ω and 6Ω, then the third resistance value must be9Ω.8Ω.6Ω.3Ω.2Ω.
You have three 1.3 kΩ resistors. A.) What is the value of the equivalent resistance for...
You have three 1.3 kΩ resistors. A.) What is the value of the equivalent resistance for the three resistors connected in series? B.) What is the value of the equivalent resistance for a combination of two resistors in series and the other resistor connected in parallel to this combination? C.) What is the value of the equivalent resistance for a combination of two resistors in parallel and the other resistor connected in series to this combination? D.) What is the...
When resistors 1 and 2 are connected in series, the equivalent resistance is 14.4 Ω. When...
When resistors 1 and 2 are connected in series, the equivalent resistance is 14.4 Ω. When they are connected in parallel, the equivalent resistance is 2.69 Ω. What are (a) the smaller resistance and (b) the larger resistance of these two resistors?
When resistors 1 and 2 are connected in series, the equivalent resistance is 22.2 Ω. When...
When resistors 1 and 2 are connected in series, the equivalent resistance is 22.2 Ω. When they are connected in parallel, the equivalent resistance is 4.39 Ω. What are (a) the smaller resistance and (b) the larger resistance of these two resistors?
What is the equivalent resistance for the circuit shown in the figure? (Figure 1) For the...
What is the equivalent resistance for the circuit shown in the figure? (Figure 1) For the circuit shown in the figure, find the current through resistor R1=6.0?(left). (Figure 1) For the circuit shown in the figure, find the potential difference across resistor R1=6.0?(left). For the circuit shown in the figure, find the current through resistor R2=15?. For the circuit shown in the figure, find the potential difference across resistor R2=15?. For the circuit shown in the figure, find the current...
Explain different ways you can arrange 4 resistors in order to get different equivalent resistance values....
Explain different ways you can arrange 4 resistors in order to get different equivalent resistance values. Mention about minimum and maximum values you can obtain.
The circuit shown in (Figure 1) contains two batteries, each with an emf and an internal resistance, and two resistors.
The circuit shown in (Figure 1) contains two batteries, each with an emf and an internal resistance, and two resistors. Part A Find the magnitude of the current in the circuit. Express your answer with the appropriate units.Part C Find the terminal voltage Vab of the 16.0 V battery. Express your answer with the appropriate units. Part D Find the potential difference Vac of point o with respect to point c. Express your answer with the appropriate units. 
Find the current through each of the three resistors of the circuit shown in the figure (Figure 1).
Problem 26.61 Find the current through each of the three resistors of the circuit shown in the figure (Figure 1). The emf sources have negligible internal resistance. Part A Find the current through 2.00-Ω resistor. Part B Find the current through 4.00-Ω resistor. Part C Find the current through 5.00 -Ω resistor.
what is the elasticity of demand between points a and b if point A is (10,54)...
what is the elasticity of demand between points a and b if point A is (10,54) and point B is (20,44)
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT