Current and Electric Circuits :: Problem 8

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Problem 8: Find the equivalent resistance of the network shown below. And also find current in 8Ω resistor if the potential difference of 12V is applied to the network.

Solution:

Step: 1 Overview

Parallel Combination of resistors.
See overview of problem 7.
Series combination of resistors.

In this combination the resistors are joined end to end providing a single path to the current between the points A and B. Due to single path flow of current in each resistor remains same, and voltage is divided in each resistor. So total voltage will be:
V = V₁+V₂+V₃+...+V_n
IR = IR₁+IR₂+IR₃+...+IR_n
R = R₁+R₂+R₃+...+R_n
Above equation is used to find equivalent resistance of resistors connected in series.

Step: 2 Calculation

From above figure, we can say that 8Ω resistor and 5Ω resistor are connected in parallel combination, so their equivalent (R_x) will be:

1/R_x=(1/8)+(1/5)
1/R_x=(5+8)/40
1/R_x=13/40
R_x=40/13
R_x=3.077Ω

After reducing the circuit, we notice that 3.077Ω resistor and 5Ω resistor are in series combination, so their equivalent (R_y) will be:

R_y=3.077+5
R_y=8.077Ω

After replacing R_y in the circuit, now R_y and 3Ω resistor are in parallel combination. So their equivalent resitance (R) will be:

1/R=1/R_y+1/3
1/R=(1/8.077)+(1/3)
1/R=11.077/24.231
R=24.231/11.077
R=2.0188Ω

Now, to find current in 8Ω resistor, we need to go back from simple to complex circuit, So starting from last circuit.

Since both R_y and 3Ω resistors are in parallel, so both will have same voltages.So voltage across R_y will be:

V_y=12V
I_y=V_y/R_y
I_y=12/8.077
I_y=1.486A

Since R_y is the series combination of R_x and 5Ω resistors, so current across R_x will be same as that of R_y. So current in R_x will be:

I_x=1.486A
V_x=I_xR_x
V_x=(1.486)(3.077)
V_x=4.57V

Finally coming to the orignal circuit, since R_x was the parallel combination of 5Ω and 8Ω resistors, so voltage across both of these resistors will be same as that of R_x. But, we are only concerned with 8Ω resistor, so voltage across 8Ω resistor will be: