A series circuit contains only a resistor and an inductor. The voltage V of the generator is fixed. If R = 20 Ω and L = 2.8 mH, find the frequency at which the current is one-fifth its value at zero frequency?
To find the frequency at which the current is one-fifth its value at zero frequency in a series circuit with only a resistor and an inductor, we can use the formula for the impedance of an inductor in a series circuit:
Z = √(R^2 + (ωL)^2)
Where Z is the impedance, R is the resistance, ω is the angular frequency, and L is the inductance.
In this case, we want to find the frequency at which the current is one-fifth its value at zero frequency. At zero frequency, the impedance of the inductor becomes zero since the reactance of an inductor is given by ωL, where ω is the angular frequency and L is the inductance. Therefore, the impedance is solely determined by the resistance (Z = R).
Let's denote the frequency at which the current is one-fifth its value at zero frequency as f. At this frequency, the impedance will be five times the resistance:
5R = √(R^2 + (2πfL)^2)
Squaring both sides of the equation yields:
25R^2 = R^2 + (2πfL)^2
Simplifying the equation, we get:
24R^2 = (2πfL)^2
Now, we can solve for the frequency, f:
f = √((24R^2) / (4π^2 L^2))
Plugging in the given values R = 20 Ω and L = 2.8 mH (convert to farads), we can calculate the frequency.