Two resistors, 6 ohms and 18 ohms are connected in parallel and the combination is connected in series with a 2 ohm resistor. Find the current through each of the first two resistors if the current through the 2 ohm resistor is 1.2 ampere. What is the voltage across each resistor?
well first the 2 ohm resistor
Vtwo = i R = 1.2*2 = 2.4 volts
Now the two in parallel
They are both at voltage V
V = i1 (6) = i2 (18)
but we know i1+i2 = 1.2
so i1 = 1.2 - i2
6(1.2- i2)=18 i2
7.2 = 24 i2
i2 = 7.2/24 = .3 amps through 18 ohm
i1 = 1.2 - .3 = 0.9 amps through 6 ohm resistor
.9*6 = .3*18 = 5.4 volts across 6 and 18 ohm resistors
To find the current through each of the first two resistors, we can start by considering the total resistance of the parallel combination.
In a parallel combination, the total resistance (R) can be calculated using the formula:
1/R = 1/R₁ + 1/R₂
Given R₁ = 6 ohms and R₂ = 18 ohms, we can substitute these values into the formula:
1/R = 1/6 + 1/18
Simplifying the equation:
1/R = (3/18) + (1/18)
1/R = 4/18
1/R = 2/9
Taking the reciprocal of both sides to find R:
R = 9/2
R = 4.5 ohms
Now, we can consider the series combination of this total resistance (4.5 ohms) with the 2 ohm resistor.
In a series combination, the total resistance (R_total) can be calculated by simply adding the resistances:
R_total = R₁ + R₂
Substituting the values, we have:
R_total = 4.5 + 2
R_total = 6.5 ohms
We know the current through the 2 ohm resistor is 1.2 amperes. By applying Ohm's Law (V = I * R), we can find the voltage across the series combination:
V = I * R_total
V = 1.2 * 6.5
V = 7.8 volts
Now, to find the current through each of the first two resistors, we can use Ohm's Law again. Since the resistors are in series, they share the same current.
Let's represent the current through both resistors as I_total. Then, we can find the voltage across each resistor using Ohm's Law.
For the 6 ohm resistor:
V₁ = I_total * R₁
6 = I_total * 6
I_total = 1 ampere
Similarly, for the 18 ohm resistor:
V₂ = I_total * R₂
18 = I_total * 18
I_total = 1/3 ampere
So, the current through the 6 ohm resistor is 1 ampere, and the current through the 18 ohm resistor is 1/3 ampere.
Now, to find the voltage across each resistor:
For the 6 ohm resistor:
V₁ = I₁ * R₁
V₁ = 1 * 6
V₁ = 6 volts
For the 18 ohm resistor:
V₂ = I₂ * R₂
V₂ = (1/3) * 18
V₂ = 6 volts
Therefore, the voltage across each resistor is 6 volts.