Current and Electric Circuits :: Problem 9

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Problem 9: A 60V potential difference is applied to the circuit shown below. Find the current in 10Ω resistor.

Solution:

Step: 1 Overview

Parallel Combination of resistors.

See overview of problem 7.

Series combination of resistors.

See overview of problem 8.

Step: 2 Calculation

parallel combination of resistors in complex circuit

10Ω and 15Ω resistors are in parallel combination, so their equivalent (R_x) will be:

1/R_x=(1/10)+(1/15)
1/R_x=(3+2)/30
1/R_x=5/30
R_x=30/5
R_x=6Ω

Now replacing R_x in the place of 10Ω and 15Ω resistors.

series combination of resistors in complex circuit

After replacing R_x in the circuit, Now R_x, and two 4Ω resistors are in series combination, so their equivalent R_y will be:

R_y=R_x+4+4
R_y=6+4+4
R_y=14Ω

Now replacing R_y in the place of three series resistors.

Now R_y and 14Ω resistors are in parallel combination, so their equivalent resistance (R_z) will be:

1/R_z=(1/R_x)+(1/14)
1/R_z=(1/14)+(1/14)
1/R_z=2/14
R_z=14/2
R_z=7Ω

Now replacing R_z in place of parallel resistors.

Now all resistors are in series, so their equivalent resistance will be:

R=R_z+5+3
R=7+5+3
R=15Ω

Now total current will be:

I=V/R
I=60V/15Ω
I=4A

Since R_z, 5Ω and 3Ω resistors are in series, so current in each resistor will be same, which is equal to total current. So current in R_z will be:

I_z=4A

And voltage across R_z will be:

V_z=I_zR_z
V_z=(4)(7)
V_z=28V

Now going back to more complex circuit:

Since R_z was the parallel combination of R_y and 14Ω resistors. So voltage across each resistor will be same as that of R_z. So voltage across R_y will be:

V_y=28V

And current across R_y will be:

I_y=V_y/R_y
I_y=28/14
I_y=2A

Now again going back to even more complex circuit consisting of R_x resistor:

R_y was the series combination of three resistors, so each resistor will have same current as that of R_y resistor. So current across R_x will be:

I_x=2A

Now voltage across R_x:

V_x=I_xR_x
V_x=(2)(6)
V_x=12V

Now coming to the orignal circuit:

Since R_x was the combination of two parallel resistors, so each will have same voltage as that of R_xresistor. So voltage across 10Ω resistor will be: