There are many different ways of solving each of these circuits. Some of you, while doing it correctly, did so in unnecessarily complicated ways.

The three steps it takes to solve each circuit are essentially the following:

• Inspect the circuit and visualize how the current flows and whether there are any branching points. Remember, current is never lost, only redistributed.

• In a series circuit, current is everywhere the same, only V and R differ. In a parallel circuit, V is same at each branching point, so only I and R can change.

• Changes in V, I and R are all governed by Ohms Law; so apply this law for a) the circuit as a whole and b) at each point there is a resistor.

Finally it is helpful to think about V,R and I, at various points in the circuit, in terms of ratios. That will be clear in the solutions that follow.

Circuit 1:

Information Given:

• The total resistance is 30
• R₄ = 10
• V₁ = 5
• V₂ = 8
• V₃ = 7

Determine values for the remaining R's and determine the total current I.

Steps:

Begin by writing down all you know:

• R₁ + R₂ + R₃ must equal 20.

• V₁ + V₂ + V₃ equals 20

• in a series circuit the current is everywhere the same so I is the same at each point in the circuit.

• Apply Ohms law to the first three resistors to get I.
  V = R x I

  V = V₁ + V₂ + V₃ =20

  R = R₁ + R₂ + R₃ =20

  Therefore I = 1 (everywhere in this circuit)

Apply Ohms law at each resistor (using I=1) and the given Voltages to determine that R₁ = 5; R₂=8; R₃=7

Circuit 2:

• Determine I₁,I₂,I₃
• Determine the total R.

First thing to notice: this is a parallel circuit. So that means

• The current through each resistor is different (the current is branching)

• The Voltage across each resistor is the same.

First you need to determine total R using the resistance in parallel rule:

• 1/R = 1/2 + 1/3 + 1/6 = 1; if 1/R = 1 then total R = 1.

• Now apply ohms law to total circuit where you know that V = 12 (as shown in the circuit) and you just determined that total R =1. V = R x I; V = 12; R= 1 therefore I = 12 amps for this circuit.

• Next apply ohms law to each resistor:

  I₁ = V/R₁ = 12/2 = 6
  I₂ = V/R₂ = 12/3 = 4
  I₃ = V/R₃ = 12/6 = 2
  

  Notice of course that I₁ + I₂ + I₃ = I = 12 as it is supposed to.

Circuit 3:

• Enter in values for I₁ (top resistor) and I₂ (bottom resistor). Total incoming current is 3 amps.

Voltage is constant in a parallel circuit. Therefore, high resistance means lower current flow through that resistor (this should be intuitive). So the current flowing through the top resistor must be less than that which is flowing through the bottom resistor.

Two ways to work this out:

Anal way:

What is total resistance:

1/R = 1/8 + 1/4 = 3/8; R = 8/3

I = 3

V = I*R = 3*8/3 so V = 8

Top resistor: V=R*I; 8=8*I; I = 1

Bottom resistor: V=R*I 8 =4*I; I = 2

Better way;

Notice the ratio of the resistors is 2:1; therefore the ratio of the currents is 1:2; that is I_top = 1 and I_bottom = 2.

Circuit 4:

• The value for all Resistors is the same, enter in values for V₁ and V₂

  This is not a strictly parallel circuit. It is a series circuit with two parts: A parallel resistance at location 2 and a series resistance at location 1. The individual R values for each resistor are identical and it matters not what the actual value of R is.

  So, for simplicity let R = 1.

  The total resistance of the parallel part is 1/R = 1/1 + 1/1 = 2 so R = 1/2.

  The series resistance is clearly R = 1.

  The same amount of current must flow through position 2 as position 1.

  The resistance at position 2 is 1/2 as much as the resistance at position 1. Therefore it takes 1/2 as much Voltage to push that current through position 2 as position 1.

  Hence V₁ = 2*V₂ and you also know that

  V₁ + V₂ = 12

  Therefore V₁ =8 and V₂ = 4.

Circuit 5:

• Under R₁,R₂, and R₃ Determine the total resistance for the configurations shown in I, II, and III
  • Circuit I: Parallel + series. The bottom total resistance is 4 + 4 = 8. The total resistance in the parallel circuit is then:

    1/R = 1/4 + 1/8 = 3/8 so R = 8/3

  • Circuit II: Again this is Parallel + Series.

    total resistance is 4 in the series part.

    Total resistance in the parallel part is:

    1/R = 1/4 + 1/4 = 1/2 so R = 2

    For the complete circuit total R = 4+2 = 6

  • Circuit III: Straight parallel circuit:

    1/R = 1/4 + 1/4 + 1/4 = 3/4 so R = 4/3