A 75.0-W lamp is placed in series with a resistor and a 120.0-V source. If the voltage across the lamp is 24 V, what is the resistance R of the resistor?
The voltage drop across the resistor is 120 - 24 = 96 V
The resistance R' of the lamp can be derived from its Wattage at 120 V:
75 W = V^2/R' = 14400/R'
R' = 192 ohms
Since the lamp gets 24/120 = 20% of the voltage drop, the other 80% of the voltage (96 V) is across the resistor.
With both in series, circuit current = lamp current = 24 V/192 ohms = 0.125 Amps
Resistor resistance = V(resistor)/I
= 96/(1/8) = 768 ohms
To find the resistance R of the resistor, we can use Ohm's Law:
Ohm's Law states that the voltage across a resistor is equal to the product of its resistance and the current passing through it:
V = I * R
Given that the voltage across the lamp is 24 V and the power of the lamp is 75.0 W, we can find the current passing through the lamp using the formula:
P = V * I
Rearranging the formula, we have:
I = P / V
Plugging in the values, we can find the current passing through the lamp:
I = 75.0 W / 24 V
Next, we need to find the total current passing through the circuit. Since the lamp and resistor are in series, the current passing through both is the same. Therefore, the total current passing through the circuit is also given by:
I = P / V
Plugging in the values, we can find the total current:
I = 75.0 W / 120.0 V
Now that we have the total current passing through the circuit, we can find the resistance R of the resistor using Ohm's Law:
R = V / I
Plugging in the values, we can find the resistance R:
R = 120.0 V / (75.0 W / 120.0 V)
Simplifying the expression:
R = 120.0 V / 1 W / 1 V
R = 120.0 V² / 1 W
Therefore, the resistance R of the resistor is 120.0 V²/W.
To find the resistance, we can apply Ohm's law. Ohm's law states that the current flowing through a resistor is directly proportional to the voltage across it and inversely proportional to its resistance.
We are given the power P of the lamp, the voltage across the lamp, and the total voltage of the circuit.
First, we can calculate the current flowing through the lamp using the power and voltage across the lamp:
P = IV
75.0 W = I * 24 V
Solving for I (current):
I = 75.0 W / 24 V
I ≈ 3.125 A
Next, we can determine the current flowing through the resistor by subtracting the current flowing through the lamp from the total current in the circuit:
I_total = V_total / R_total
Since the lamp and resistor are in series, the current through both must be the same:
I_total = I + I_resistor
Substituting the values we have:
3.125 A = 3.125 A + I_resistor
Solving for I_resistor:
I_resistor = 3.125 A - 3.125 A
I_resistor = 0 A
From this, we can conclude that the current flowing through the resistor is zero ampere (0 A).
Finally, we can use Ohm's law to find the resistance R of the resistor:
V_resistor = I_resistor * R
Since the current is 0 A, the voltage across the resistor is also 0 V:
0 V = 0 A * R
Therefore, the resistance of the resistor is 0 ohms (Ω).
In summary, the resistance R of the resistor in this circuit is 0 ohms (Ω).
Vr = 120-24 = 96 V. = Voltage across the
resistor.
V*I = 75W.
24I = 75
I = 3.125A = Lamp current
R = Vr/I = 96/3.125 = 30.72 Ohms in series with lamp.