Question

A generator is connected to a resistor and a 0.032-H inductor in series. The rms voltage across the generator is 7.2 V. When the generator frequency is set to 110 Hz, the rms voltage across the inductor is 3.7 V. Determine the resistance of the resistor in this circuit.

Answer 1

The resistance of the resistor in this RLC circuit can be calculated using the impedance formula and the given values of the rms voltage and inductance. The resistance is approximately 4.91 Ω.

Step-by-step explanation:

In an RLC circuit, the rms voltage across the inductor can be calculated using the formula:

VL = IL * XL

Where VL is the voltage across the inductor, IL is the current through the inductor, and XL is the reactance of the inductor. The reactance of the inductor can be calculated using the formula:

XL = 2πfL

Where f is the frequency of the generator and L is the inductance of the inductor.

In this case, the rms voltage across the inductor is given as 3.7 V and the frequency is given as 110 Hz. Plugging in these values into the formula, we can calculate XL as follows:

XL = 2π * 110 * 0.032

Simplifying, we get:

XL ≈ 21.76 Ω

Now, since the resistor and the inductor are connected in series, the total impedance of the circuit can be calculated as follows:

Z = √(R2 + XL2)

Given that the rms voltage across the generator is 7.2 V, we can also use this formula to calculate the resistance of the resistor:

R = √(Z2 - XL2)

By plugging in the values of Z and XL, we can find the resistance R:

R = √((7.22) - (21.762))

Simplifying, we get:

R ≈ 4.91 Ω

Therefore, the resistance of the resistor in this circuit is approximately 4.91 Ω.

Answer 2

The resistance of the resistor in the circuit can be calculated using Ohm’s Law and the impedance of the circuit. The rms voltage across the resistor is calculated by subtracting the rms voltage across the inductor from the total rms voltage across the circuit. The resistance R can then be found by dividing the voltage across the resistor by the rms current through the circuit.

Step-by-step explanation:

The rms voltage across the generator is 7.2 V and the rms voltage across the inductor is 3.7 V. Since the resistor and inductor are connected in series, the total rms voltage across the circuit is the sum of the voltage across the resistor and the voltage across the inductor. Therefore, we can write the following equation:

7.2 V = VR + 3.7 V

Solving for VR, we get:

VR = 7.2 V - 3.7 V = 3.5 V

The resistance of the resistor can be calculated using Ohm's Law:

R = VR / I

where I is the rms current through the circuit. The rms current can be found using the formula:

I = Vgenerator / Z

where Z is the impedance of the circuit. The impedance of a series RLC circuit is given by:

Z = sqrt((R2 + (wL - 1/(wC))²))

where w is the angular frequency given by 2πf. Plugging in the values, we can solve for Z and then substitute it back to find the resistance R.