Resistance: Electrical Circuits in Parallel

  This problem set deals with electrical circuits in parallel.  
1. Two resistors are connected in parallel as shown. Find the total resistance of the circuit.

 

 
Knowns Unknowns Equations/Remarks
R1 = 1 W
R2 = 4 W 		Rtot = ? 1/Rtot = 1/R1+ 1/R2 
1/Rtot = 1/R1+ 1/R2 = 1/(1 W) + 1/(4 W) = 5/(4 W)

Rtot = 0.8 W

Substitute in the known values and calculate the answer.
2. Copper wire and solder are each classified as conductors.  
3. Given the circuit shown:
  1. Find the total resistance, Rtot.
  2. Find the total current, Itot.
  3. Find voltage across each resistor.
  4. Find the current through each resistor.
 
     
Knowns Unknowns Equations/Remarks
Vtot = 10 V
R1 = 15 W
R2 = 45 W 		a. Rtot = ?
b. Itot = ?
c. V1 and V2 = ?
d. I1 and I2 = ?		 1/Rtot = 1/R1+ 1/R2 
Rtot = Vtot/Itot 
R = V/I 
a. 1/Rtot = 1/R1+ 1/R2 = 1/(15 W) + 1/(45 W)
    Rtot = 11.25 W

b. Rtot = Vtot/Itot 
    Itot = Vtot/Rtot = (10 V)/(11.25 W)
    Itot = 0.889 A

c. Vtot = V1 = V2 = 10 V

d. R = V/I
    I1 = V1/R1 = (10 V)/(15 W) = 0.667 A

    I2 = V2/R2 = (10 V)/(45 W) = 0.222 A

Substitute in the known values and calculate the answer.

Use the total voltage and total resistance to find the total current.

Use the voltage drop across each resistor along with its resistance to find the current going through each resistor.
4. Given a parallel circuit with a 20 V battery and three resistors in parallel: R1 = 5 W, R2 = 8 W, and R3 = 10 W. Find the current through each resistor, the total current, Rtot using Ohm's Law and Rtot by calculating it for the parallel circuit.

  
Knowns Unknowns Equations/Remarks
Vtot = 20 V
R1 = 5 W
R2 = 8 W
R3 = 10 W
V1=V2=V3=Vtot 		I1 = ?
I2 = ?
I3 = ?
Itot = ?
Rtot = ? by Ohm's Law
Rtot = ? by parallel circuit		 R = V/I 
Itot = I1 + I2 + I3 
Rtot = Vtot/Itot 
1/Rtot = 1/R1+ 1/R2 + 1/R3 
R1 = V1/I1 
I1 = V1/R1 = (20 V)/(5 W) = 4 A

R2 = V2/I2 
I2 = V2/R2 =  (20 V)/(8 W) = 2.5 A

R3 = V3/I3 
I3 = V3/R3 =  (20 V)/(10 W) = 2 A

Itot = I1 + I2 + I3 = 4 A + 2.5 A + 2 A
Itot = 8.5 A

Rtot = Vtot/Itot = (20 V)/(8.5 A)
Rtot = 2.35 W

1/Rtot = 1/R1+ 1/R2 + 1/R3 
1/Rtot = 1/(5 W)+ 1/(8 W) + 1/(10 W) 
Rtot = 2.35 W

Find the current going through each resistor by rearranging the equation to solve for current, substitute in the known values, and calculate the answer.

Find the total current by adding up the currents flowing through the resistors.

Find the total resistance in the circuit by using Ohm's Law and the total voltage and total current.

Find the total resistance by calculating it with the parallel circuit resistance equation.
5.
Given the circuit shown,
  1. find the current in each leg,
  2. find the total current, and
  3. find the total resistance two different ways.
 
     
Knowns Unknowns Equations/Remarks
Vtot = 12 V
R1 = 27 W
R2 = 15 W
R3 = 18 W
V1=V2=V3=Vtot 		a. I1 = ?
    I2 = ?
    I3 = ?
b. Itot = ?
c. Rtot = ? by Ohm's Law
    Rtot = ? by parallel circuit		 R = V/I 
Itot = I1 + I2 + I3 
Rtot = Vtot/Itot 
1/Rtot = 1/R1+ 1/R2 + 1/R3 
a. R1 = V1/I1 
    I1 = V1/R1 = (12 V)/(27 W) = 0.444 A

    R2 = V2/I2 
    I2 = V2/R2 =  (12 V)/(15 W) = 0.8 A

    R3 = V3/I3 
    I3 = V3/R3 =  (12 V)/(18 W) = 0.667 A

b. Itot = I1 + I2 + I3 = 0.444 A + 0.8 A + 0.667 A
    Itot = 1.91 A

c. Rtot = Vtot/Itot = (12 V)/(1.91 A)
    Rtot = 6.28 W

    1/Rtot = 1/R1+ 1/R2 + 1/R3 
    1/Rtot = 1/(27 W)+ 1/(15 W) + 1/(18 W) 
    Rtot = 6.28 W

Find the current going through each resistor by rearranging the equation to solve for current, substitute in the known values, and calculate the answer.

Find the total current by adding up the currents flowing through the resistors.

Find the total resistance in the circuit by using Ohm's Law and the total voltage and total current.

Find the total resistance by calculating it with the parallel circuit resistance equation.
6. What is the total resistance across two resistors in parallel if their values are 4 W and 8 W?

  
Knowns Unknowns Equations/Remarks
R1 = 4 W
R2 = 8 W 		Rtot = ? 1/Rtot = 1/R1+ 1/R2 
1/Rtot = 1/R1+ 1/R2 = 1/(4 W) + 1/(8 W)

Rtot = 2.67 W

Substitute in the known values and calculate the answer.