A 120-V rms voltage at 1000 Hz is applied to a resistor and an inductor in series. If the impedance of this circuit is 110 Ù, what is the maximum value of the current?
I = V/Z = 120 / 110 = 1.09Amps.
To find the maximum value of the current in the circuit, we need to use Ohm's Law.
Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage across the two points, and inversely proportional to the resistance.
In this case, since we have a resistor and an inductor in series, the impedance (Z) of the circuit is a combination of the resistance (R) and the reactance (X) of the inductor.
The reactance (X) of an inductor is given by the formula X = 2πfL, where f is the frequency and L is the inductance.
So, in this case, the impedance can be calculated as: Z = R + jX, where j is the imaginary unit.
Given that the impedance (Z) is 110 Ω, the rms voltage (V) is 120 V, and the frequency (f) is 1000 Hz, we can write the equation as:
110 = R + j(2πfL)
Since we are given the impedance (Z) and not the resistance (R), we need to find the value of the reactance (X) of the inductor from the impedance equation.
To do this, let's rearrange the equation:
j(2πfL) = 110 - R
The magnitude of the reactance (|X|) can be calculated using the equation |X| = √(Re(X)^2 + Im(X)^2).
Since we know that the magnitude of the impedance is given by Z = √(Re(Z)^2 + Im(Z)^2), and the imaginary part of the impedance (Im(Z)) is equal to the magnitude of the reactance (|X|), we can rewrite the equation:
110 = √(R^2 + |X|^2)
Simplifying the equation, we have:
12100 = R^2 + |X|^2
Now, we can substitute the reactance (X) in terms of frequency (f) and inductance (L):
12100 = R^2 + (2πfL)^2
Solving for the reactance, we have:
(2πfL)^2 = 12100 - R^2
(2πfL)^2 = 12100 - R^2
(2πfL)^2 = 12100 - R^2
(2πfL)^2 - 12100 = -R^2
R^2 = 12100 - (2πfL)^2
R = √(12100 - (2πfL)^2)
Once we know the value of the resistance (R), we can use Ohm's Law to calculate the current (I):
I = V / Z
where V is the rms voltage.
Substituting the known values, we have:
I = 120 / (R + j(2πfL))
Given all the values, you can now calculate the maximum value of the current in the circuit by substituting the values of R and L, and then calculating the real part of the complex impedance.