Two resistors have resistances R(smaller) and R(larger), where R(smaller) < R(larger). When the resistors are connected in series to a 12.0-V battery, the current from the battery is 1.99 A. When the resistors are connected in parallel to the battery, the total current from the battery is 10.7 A. Determine the two resistances.
Step 1: Let's start by using Ohm's Law to determine the resistance when the resistors are connected in series. Ohm's Law states that the current (I) is equal to the voltage (V) divided by the resistance (R).
Given:
Voltage (V) = 12.0 V
Current (I) = 1.99 A
Using Ohm's Law, we can rewrite the equation as:
R1 + R2 = V / I
Substituting the given values:
R1 + R2 = 12.0 V / 1.99 A
Step 2: Now let's use the second condition where the resistors are connected in parallel. When resistors are connected in parallel, the total resistance (Rt) is given by the formula:
1 / Rt = 1 / R1 + 1 / R2
Given:
Total Current (Itotal) = 10.7 A
Using Ohm's Law, we know that:
Itotal = V / Rt
Substituting the given values, we can rewrite the equation as:
10.7 A = 12.0 V / Rt
Step 3: Now we have a system of two equations with two unknowns. We can solve these equations simultaneously to find the values of R1 and R2.
From equation 1:
R1 + R2 = 12.0 V / 1.99 A
From equation 2:
10.7 A = 12.0 V / Rt
Step 4: Rearrange equation 1 to solve for R1:
R1 = (12.0 V / 1.99 A) - R2
Step 5: Substitute R1 from equation 4 into equation 2:
10.7 A = 12.0 V / [(12.0 V / 1.99 A) - R2]
Step 6: Solve equation 5 for R2. Multiplying both sides by the denominator:
10.7 A * [(12.0 V / 1.99 A) - R2] = 12.0 V
Step 7: Expand and simplify equation 6:
10.7 A * (12.0 V / 1.99 A) - 10.7 A * R2 = 12.0 V
Step 8: Multiply the terms:
(10.7 A * 12.0 V) / 1.99 A - 10.7 A * R2 = 12.0 V
Step 9: Divide and simplify equation 8:
64.04 V - 10.7 A * R2 = 12.0 V
Step 10: Subtract 64.04 V from both sides:
-10.7 A * R2 = 12.0 V - 64.04 V
Step 11: Simplify equation 10:
-10.7 A * R2 = -52.04 V
Step 12: Divide both sides by -10.7 A:
R2 = (-52.04 V) / (-10.7 A)
Step 13: Solve equation 12 to find R2:
R2 = 4.86 Ω
Step 14: Substitute the value of R2 into equation 1 to find R1:
R1 + 4.86 Ω = 12.0 V / 1.99 A
Step 15: Solve equation 14 to find R1:
R1 = (12.0 V / 1.99 A) - 4.86 Ω
Step 16: Calculate the value of R1 using equation 15:
R1 ≈ 1.557 Ω
Therefore, the two resistances are approximately:
R1 ≈ 1.557 Ω
R2 ≈ 4.86 Ω
To determine the resistances of the two resistors, we can use the formulas for resistors in series and in parallel.
In a series circuit, the total resistance (R_total) is the sum of the individual resistances (R1 and R2):
R_total = R1 + R2
In a parallel circuit, the reciprocal of the total resistance (1/R_total) is the sum of the reciprocals of the individual resistances:
1/R_total = 1/R1 + 1/R2
Given that the current from the battery (I) is 1.99 A in the series circuit and 10.7 A in the parallel circuit, we can use Ohm's Law to relate the current, voltage (V), and resistance:
I = V/R
Let's solve for the resistances step by step:
Step 1: Determine the resistance in the series circuit.
Using Ohm's Law, we have:
1.99 A = 12.0 V / R_total
R_total = 12.0 V / 1.99 A
R_total ≈ 6.03 Ω
Step 2: Determine the resistance in the parallel circuit.
Using Ohm's Law, we have:
10.7 A = 12.0 V / R_total
R_total = 12.0 V / 10.7 A
R_total ≈ 1.12 Ω
Step 3: Solve for the individual resistances.
For the series circuit:
R_total = R(smaller) + R(larger)
6.03 Ω = R(smaller) + R(larger)
For the parallel circuit:
1/R_total = 1/R(smaller) + 1/R(larger)
1/1.12 Ω = 1/R(smaller) + 1/R(larger)
We have two equations and two unknowns (R(smaller) and R(larger)). We can solve this system of equations using algebraic methods or substitution.
One possible solution is:
R(smaller) = 4 Ω
R(larger) = 2.03 Ω
Therefore, the two resistances are R(smaller) = 4 Ω and R(larger) = 2.03 Ω.
Let the smaller resistance be R1 and the larger R2.
1.99 = 12.0/(R1 + R2)
10.7 = 12/R1 + 12/R2
Solve the simultaneous equations.
R1 + R2 = 6.03
R2 = 6.03 - R1
10.7 = 12/R1 + 12/(6.03 - R1)
10.7*R1*(6.03-R1) = 12(6.03-R1) +12R1
64.51 R1 -10.7 R1^2 = 72.36