Answer:
R(smaller) = 1.3 Ω and R(larger) = 5.4 Ω
Explanation:
Ohm's Law states that:
V = IR
R = V/I
where,
R = Resistance
V = Potential Difference
I = Current
Therefore, for series connection:
Rs = Vs/Is
where,
Rs = Resistance when connected in series = R(smaller) + R(larger)
Vs = Potential Difference when connected in series = 12 V
Is = Current when connected in series = 1.78 A
Therefore,
R(smaller) + R(larger) = 12 V/1.78 A
R(smaller) + R(larger) = 6.74 Ω --------------- equation 1
R(smaller) = 6.74 Ω - R(larger) --------------- equation 2
Therefore, for series connection:
Rp = Vp/Ip
where,
Rp = Resistance when connected in parallel = [1/R(smaller) + 1/R(larger)]⁻¹
Rp = [{R(smaller) + R(larger)}/{R(smaller).R(larger)]⁻¹
Rp = R(smaller).R(larger)/[R(smaller) + R(larger)]
Vp = Potential Difference when connected in parallel = 12 V
Ip = Current when connected in parallel = 11.3 A
Therefore,
R(smaller).R(larger)/[R(smaller) + R(larger)] = 12 V/11.3 A
using equation 1 and equation 2, we get:
[6.74 Ω - R(larger)].R(larger)/6.74 Ω = 1.06 Ω
6.74 R(larger) - R(larger)² = (6.74)(1.06)
R(larger)² - 6.74 R(larger) + 7.16 = 0
solving this quadratic equation we get:
R(larger) = 5.4 Ω (OR) R(larger) = 1.3 Ω
using these values in equation 2, we get:
R(smaller) = 1.3 Ω (OR) R(smaller) = 5.4 Ω
Since, it is given in the question that R(smaller)<R(larger).
Therefore, the correct answers will be:
R(smaller) = 1.3 Ω and R(larger) = 5.4 Ω
