Question

From the circuit as shown in the diagram, please provide a detailed procedure for finding the current flowing through R3 in terms of how many times the current flowing through R1, which is denoted as I. All resistors in the circuit have the same resistance value, denoted as R, and V1 has a value of V.

Answer 1

R2-R5 series current
`(I_s)`=
`3V1 / 2R`. Since all series resistors have equal
`R, I_s =` current throug
`h R3 (I_R3)`= 3 times current through
`R1 (V1/R).`

Certainly! Here's how to find the current flowing through
`R3 (I_R3)\\` in terms of the current flowing through
`R1 (I_R1)` in the circuit you provided:

1. Identify the series and parallel connections:

* Notice that R2, R3, R4, and R5 are connected in series. Let the current flowing through this series combination be
`I_s.`

* Similarly, R8, R9, R10, and R11 are connected in series. Let the current flowing through this series combination be
`I_t.`

* R6 and R7 are connected in parallel.

* R1, R12, and R13 are connected in parallel.

2. Apply Kirchhoff's Current Law (KCL) at the junction of R2, R6, and R1:

* At the junction where R2, R6, and R1 meet, the total incoming current must equal the total outgoing current.

* Therefore,
`I_R1 + I_R6 = I_s`

3. Apply Ohm's Law to R1 and R6:

*
`I_R1 = V1 / R`(since R1 is directly connected to V1)

*
`I_R6 = V1 / (2R)`(because R6 and R7 are in parallel, their combined resistance is R/2)

4. Substitute the expressions for I_R1 and I_R6 from step 3 into the equation from step 2:

`* V1 / R + V1 / (2R) = I_s`

* Combine like terms:
`3V1 / (2R) = I_s`

5. Analyze the series connection of R2, R3, R4, and R5:

* Since all resistors in the series combination have the same resistance (R), the current will be the same throughout the series. Therefore,
`I_s = I_R3.`

6. Express I_R3 in terms of I_R1:

* Substitute the expression for I_s from step 4 into the equation from step 5:

`* 3V1 / (2R) = I_R3`

* Therefore,
`I_R3 = (3V1) / (2R)`

Conclusion:

The current flowing through R3
`(I_R3)` is 3 times the current flowing through
`R1 (I_R1)` and is equal to
`(3V1) / (2R).`

Note: This procedure assumes that the circuit is in steady state and all the resistors are linear.