Three resistors R₁, R₂, and R, are connected in series and to a 120-volt source. If R₂ = 2R₁, R3 = 3R₁,
and the total power taken by the circuit is 200 watts, calculate (a) the resistance of each resistor, (b) the power in each resistor.
Total resistance R = 6R₁
P = E^2/R, so
120^2/6R₁ = 200
solve for R₁
the current is 120/6R₁, so use that for each resistor, P = I^2 R
P₁ = 120/6R₁ * R₁ = 20
and so on for the other two resistors
12
(a) Well, since the resistors are connected in series, their equivalent resistance (R) can be calculated by adding them up. So, we have R = R₁ + R₂ + R₃ = R₁ + 2R₁ + 3R₁ = 6R₁.
Now, since we know the total power (P) drawn by the circuit and the voltage (V) across it, we can use the formula P = V² / R to find the value of each resistor.
We have P = 200 watts and V = 120 volts. Plugging these values into the formula, we get:
200 = 120² / (6R₁)
200 = 14400 / (6R₁)
6R₁ = 14400 / 200
6R₁ = 72
R₁ = 72 / 6
R₁ = 12 ohms
Now that we have the value of R₁, we can find the values of R₂ and R₃ using the given relationships:
R₂ = 2R₁ = 2 * 12 = 24 ohms
R₃ = 3R₁ = 3 * 12 = 36 ohms
So, the resistance of each resistor is R₁ = 12 ohms, R₂ = 24 ohms, and R₃ = 36 ohms.
(b) To find the power in each resistor, we can use the formula P = V² / R.
For R₁, we have P₁ = 120² / 12 = 1200 watts
For R₂, we have P₂ = 120² / 24 = 600 watts
For R₃, we have P₃ = 120² / 36 = 400 watts
So, the power in each resistor is P₁ = 1200 watts for R₁, P₂ = 600 watts for R₂, and P₃ = 400 watts for R₃.
To solve this problem, we'll follow these steps:
Step 1: Determine the relationship between the resistors.
Step 2: Use the relationship to express the resistance of each resistor in terms of a common variable.
Step 3: Use the given information to form an equation representing the total power.
Step 4: Solve the equation to find the value of the common variable.
Step 5: Calculate the resistance of each resistor and the power in each resistor.
Let's start!
Step 1: Determine the relationship between the resistors.
Given:
R₂ = 2R₁
R₃ = 3R₁
Step 2: Express the resistance of each resistor using a common variable.
Let's use R₁ as the common variable. So:
R₂ = 2R₁
R₃ = 3R₁
Step 3: Form an equation representing the total power.
Power (P) is given by the formula: P = V^2 / R,
where V is the voltage and R is the resistance. In this case, the total power is given as 200 watts, and the voltage is 120 volts. Therefore:
P_total = 200 W = (120 V)^2 / (R₁ + R₂ + R₃)
Step 4: Solve the equation to find the value of the common variable.
Substituting the expressions for R₂ and R₃ from step 2 into the equation from step 3:
200 = (120^2) / (R₁ + 2R₁ + 3R₁)
200 = 14400 / 6R₁
6R₁ = 14400 / 200
R₁ = 14400 / (200 * 6)
R₁ = 12 Ω
Step 5: Calculate the resistance of each resistor and the power in each resistor.
(a) Resistance of each resistor:
R₁ = 12 Ω (already found in step 4)
R₂ = 2R₁ = 2 * 12 Ω = 24 Ω (using the expression from step 2)
R₃ = 3R₁ = 3 * 12 Ω = 36 Ω (using the expression from step 2)
(b) Power in each resistor:
Using the formula P = V^2 / R:
Power in R₁ = (120 V)^2 / 12 Ω = 1200 W
Power in R₂ = (120 V)^2 / 24 Ω = 600 W
Power in R₃ = (120 V)^2 / 36 Ω = 400 W
So, the resistance of each resistor is:
(a) R₁ = 12 Ω
R₂ = 24 Ω
R₃ = 36 Ω
And the power in each resistor is:
(b) Power in R₁ = 1200 W
Power in R₂ = 600 W
Power in R₃ = 400 W
To solve this problem, we will use Ohm's law and the formula for power in a resistor.
(a) Resistance of each resistor:
Since the resistors are connected in series, the total resistance in the circuit is the sum of the individual resistances:
R_total = R₁ + R₂ + R₃
Substituting the given values:
R_total = R₁ + 2R₁ + 3R₁
R_total = 6R₁
To find the resistance of each resistor, we can divide the total resistance by the appropriate factor:
R₁ = R_total / 6
R₂ = 2R_total / 6 = R_total / 3
R₃ = 3R_total / 6 = R_total / 2
(b) Power in each resistor:
The power consumed by a resistor can be calculated using the formula:
P = V² / R
where P is the power in watts, V is the voltage across the resistor in volts, and R is the resistance in ohms.
Since all resistors are connected in series, the same current passes through each resistor, and the voltage across each resistor is proportional to its resistance:
V₁ = R₁ * I
V₂ = R₂ * I
V₃ = R₃ * I
Substituting the given values:
V₁ = (R_total / 6) * I
V₂ = (R_total / 3) * I
V₃ = (R_total / 2) * I
Now we can calculate the power in each resistor:
P₁ = (V₁)² / R₁ = [(R_total / 6) * I]² / R₁
P₂ = (V₂)² / R₂ = [(R_total / 3) * I]² / R₂
P₃ = (V₃)² / R₃ = [(R_total / 2) * I]² / R₃
Since the total power consumed by the circuit is given as 200 watts, we can sum the powers in each resistor:
P_total = P₁ + P₂ + P₃
Substituting the formulas for each power, we can solve for the current, I:
P_total = [(R_total / 6) * I]² / R₁ + [(R_total / 3) * I]² / R₂ + [(R_total / 2) * I]² / R₃
Solving this equation will give us the current, I.
Once we have the current, we can substitute it back into the formulas for each power to calculate the power in each resistor.