What is the resonant frequency of
the circuit shown in the diagram?
The circuit consists of an
alternating-voltage source connected to a series combination of a 35-Ω resistor, a
7.5-henry inductor, and a 350-microfarad capacitor. And we’re asked to find the
resonant frequency of this circuit. Recall that the inductive reactance
in a circuit is the angular frequency of the voltage source times the
inductance. And the capacitive reactance is one
divided by the angular frequency of the voltage source times the capacitance. On resonance, these two reactances
If we call the resonant angular
frequency 𝜔 naught, then we have that 𝜔 naught 𝐿 is equal to one over 𝜔 naught
𝐶, which we can solve for 𝜔 naught. When we solve this equation for 𝜔
naught, we find that the resonant angular frequency is equal to one divided by the
square root of the inductance of the inductor times the capacitance of the
capacitor. Now, this is a formula for angular
frequency, but we’re looking for just regular frequency. So we need to use the relationship
that angular frequency is two 𝜋 times the regular frequency.
Alright, so let’s plug our
definition for angular frequency into our equation for the resonant angular
frequency. We have two times 𝜋 times the
resonant frequency is equal to one divided by the square root of the inductance
times the capacitance. To get this expression into the
final form we need, we simply divide both sides by two 𝜋. On the left-hand side, two 𝜋
divided by two 𝜋 is one, and we’re just left with 𝑓 naught. On the right-hand side, the two 𝜋
just becomes part of the denominator of our fraction. This leaves us with the final
formula we need. Resonant frequency is equal to one
divided by two 𝜋 times the square root of the inductance, in henries, times the
capacitance, in farads.
So now we just need to plug in
values. We have an inductance in
henries. It’s 7.5 henries. However, our capacitance is given
in microfarads instead of farads. To convert to farads, recall that
there are one million microfarads per farad. In other words, one microfarad is
equivalent to 10 to the negative sixth farads. Since we have 350 microfarads, our
capacitance is equivalent to 350 times 10 to the negative sixth farads.
Plugging our inductance and
capacitance into our formula for resonant frequency, this gives us one divided by
two 𝜋 times the square root of 350 times 10 to the negative sixth farads times 7.5
henries. It turns out that the square root
of one farad times one henry is one second. So we can rewrite the denominator
with units of seconds.
Now, one divided by seconds is the
unit hertz, which is used for frequency. So now we have an expression for
the resonant frequency. That’s a number times a unit hertz,
which is the right unit for frequency. So now all we have to do is
evaluate this number with a calculator. When we do this evaluation, we find
that the entire numerical expression is approximately equal to 3.1. So the resonant frequency of this
circuit is 3.1 hertz. It’s worth noting that the 35-Ω
resistor played no role in our calculation of the resonant frequency.