A series circuit consists of two 12 V batteries and a motor, through which a current of 4 A is flowing. Calculate the resistance of the motor.

The total voltage to which the motor is connected is the sum of the voltages of the two batteries, when connected in series: V = V ₁ + V ₂ = Cloze (1):JXUwMDY5JXUwMDAz + Cloze (2):JXUwMDY5JXUwMDAz = 24 Cloze (3):JXUwMDBl

According to Ohm's law: R = V / I = Cloze (4):JXUwMDZhJXUwMDA2 / Cloze (5):JXUwMDZj = 6 Ω

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Known quantities:  R₁ = 30 Ω,  R₂ = Cloze (1):JXUwMDY5JXUwMDA0   Ω,  V = 4.5 V

Unknown quantities : R, I

Formulas:

Total or equivalent resistance of the series circuit:  R = R₁ + R₂

Ohm´s Law:  V = Cloze (2):JXUwMDEx · R

We want to calculate I, so we rearrange Ohm´s Law to become: 

I = Cloze (3):JXUwMDBl Cloze (4):JXUwMDc3 Cloze (5):JXUwMDBh

Plug in the values:

R = 30 + 15 = 45 Ω

I = V / R = Cloze (6):JXUwMDZjJXUwMDFhJXUwMDFi  / Cloze (7):JXUwMDZjJXUwMDAx = 0.1 Cloze (8):JXUwMDE5

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Known quantities:  R₁ = 0.04 kΩ = Cloze (1):JXUwMDZjJXUwMDA0  Ω,  Cloze (2):JXUwMDBh ₂ =  Cloze (3):JXUwMDZlJXUwMDA2  Ω,  Cloze (4):JXUwMDBl = Cloze (5):JXUwMDYx V                            

Unknowns:    a) V₁ and Cloze (6):JXUwMDEx        b) V₂

Series circuit  formulas:

Cloze (7):JXUwMDBh = R₁ + R₂                        V = I · R                        V₁ = I · R₁                     V₂ = I · R₂

Rearrange the formulas to calculate the unknowns:   I =  Cloze (8):JXUwMDBl  /  Cloze (9):JXUwMDBh

Plug in our values:

a) R = 40 + Cloze (10):JXUwMDZlJXUwMDA2 = 100 Ω

I = V / R = Cloze (11):JXUwMDYx / Cloze (12):JXUwMDY5JXUwMDAxJXUwMDAw = 0.09 Cloze (13):JXUwMDE5

Notice the order in which we have solved the problem even though the first thing they ask us is V₁ :

1. We calculated R because it was necessary to calculate l.

2. We calculated I because it was necessary to calculate V₁ and V₂

3. NOW we can calculate the potential difference between the ends of the 40 Ω resistor (V₁):

V₁ = I · R₁ = Cloze (14):JXUwMDY4JXUwMDFlJXUwMDFlJXUwMDA5 · Cloze (15):JXUwMDZjJXUwMDA0 = 3.6 V

b) In the other resistance:  V₂ = I · R₂ = Cloze (16):JXUwMDY4JXUwMDFlJXUwMDFlJXUwMDA5 · Cloze (17):JXUwMDZlJXUwMDA2 = 5.4 V

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