### Video Transcript

If the π factor of a RLC circuit
is calculated using the formula π equals one over π
square root πΏ over πΆ,
calculate the π factor of a circuit that contains a 555-millihenry inductor and a
32.4-kiloohm resistor if the resonant frequency of the circuit is 247 kilohertz. Give your answer to one decimal
place.

In this question, we are asked to
calculate π factor of a RLC circuit using the equation given to us.

Letβs say that this is the circuit
weβre working with. It has a resistor, inductor, and
capacitor. And since this is an alternating
current circuit, it has a variable voltage supply. We are given the resistance π
of
the resistor and the inductance πΏ of the inductor. But the value of the capacitance πΆ
is not stated in the question. To determine the capacitance, we
will recall an equation that can relate πΆ to our other variables.

The resonant frequency π of a
circuit is given by the equation two ππ equals the square root of one over πΏπΆ,
where πΏ is the inductance of the circuit and πΆ is the capacitance of the
circuit. We now need to make πΆ the
subject. We can do this by squaring both
sides, which gives two ππ squared equals one over πΏπΆ. We can then take the reciprocal of
both sides to give us one over two ππ squared equals πΏπΆ. And finally, we can divide both
sides by πΏ to leave us with πΆ equals one over two ππ squared πΏ.

We can now substitute this equation
for the capacitance πΆ into the equation of the π factor that we were given in the
question to get π equals one over π
multiplied by the square root of two ππ
squared πΏ squared. The square root will cancel the
squared terms, leaving us with π equals two πππΏ over π
.

Before we substitute in the given
variables, we need to be careful of the unit prefixes. The resonant frequency is given as
247 kilohertz, which is equal to 247 times 10 to the power three hertz. The inductance πΏ is given as 555
millihenries, which is equal to 555 times 10 to the power negative three
henries. And the resistance π
is given as
32.4 kiloohms, which is equal to 32.4 times 10 to the power three ohms.

Now then, if we substitute in our
given variables into this equation, we find that π is equal to two π multiplied by
247 times 10 to the power three hertz multiplied by 555 times 10 to the power
negative three henries over 32.4 times 10 to the power three ohms. Completing this calculation, we
find that π is equal to 26.6 when rounded to one decimal place, which is a
dimensionless number.

So, the π factor for RLC circuit
with a 555-millihenry inductor, a 32.4-kiloohm resistor, and a resonant frequency of
247 kilohertz is 26.6.