Question

equivalent resistance
two resistors of resistance ( r_3 = 3.00 omega ) and ( r_4 = 3.00 omega ) are added to the network, and an additional resistor of resistance ( r_3 = 3.00 omega ) is connected by a switch, as shown in the diagram. find the equivalent resistance
( r_{eq} ) of the new resistor network when the switch is open.
express your answer in ohms.
view available hint(s)

all attempts used; correct answer displayed

part c
find the equivalent resistance ( r_{eq} ) of the resistor network described in part b when the switch is closed.
express your answer in ohms.
view available hint(s)

Explanation:

Step1: Analyze the circuit when the switch is closed

When the switch is closed, \(R_3\) and \(R_4\) are short - circuited (current takes the path of least resistance through the switch).

Step2: Calculate the equivalent resistance of \(R_1\) and \(R_2\) in parallel

The formula for two resistors \(R_a\) and \(R_b\) in parallel is \(R_{eq}=\frac{R_a\times R_b}{R_a + R_b}\). Here \(R_1 = R_2=3.00\space\Omega\), so \(R_{12}=\frac{3\times3}{3 + 3}=\frac{9}{6}=1.5\space\Omega\)

Step3: Calculate the equivalent resistance of \(R_{12}\) in series with \(R_5\) and \(R_6\)

The formula for resistors in series is \(R_{total}=R_1+R_2+\cdots+R_n\). Here \(R_5 = R_6 = 3.00\space\Omega\) and \(R_{12}=1.5\space\Omega\). So \(R_C=R_{12}+R_5+R_6\)
Substitute the values: \(R_C=1.5 + 3+3\)

Answer:

\(R_C = 7.5\space\Omega\)