Video Transcript
In the following RLC circuit, if
the ammeter reads a current of 5 amperes, calculate the value of the capacitor’s
capacitance. Give your answer in scientific
notation to two decimal places.
Here, we’ve been given an RLC
circuit diagram and information about its various components. Our job is to calculate the unknown
capacitance of the capacitor, labeled as 𝐶. To do this, we can recall a formula
that relates capacitance, 𝐶, to the resonant frequency, 𝐹, and inductance, 𝐿, of
a circuit: two 𝜋𝐹 equals the square root of one over 𝐿𝐶.
Since we’re interested in solving
for 𝐶, let’s rearrange this equation to make 𝐶 the subject. We can start by squaring both sides
to undo the radical that 𝐶 appears under. Then, we’ll take the reciprocal of
both sides to move 𝐶 from the denominator to the numerator. Finally, we can divide both sides
by 𝐿 to get 𝐶 by itself. Thus, the expression can be written
as 𝐶 equals one over two 𝜋𝐹 squared times 𝐿.
Since we already know the values of
inductance and resonant frequency for this circuit, we have everything we need to
find the capacitance 𝐶. Notice that we don’t even need to
use all of the values given to us in the circuit diagram, and that’s okay. We know that 𝐹 equals 100 hertz
and that 𝐿 equals three henries. Both of these values are already
expressed in SI-derived units, so they’re ready to be substituted into the
formula.
Doing this and grabbing a
calculator, we get a result of 8.4434 and so on times 10 to the negative seven
farads, which is the SI-derived unit of capacitance. We’ve been told to give our answer
in scientific notation to two decimal places. This is already in scientific
notation. So just rounding to two decimal
places, we reach a final answer of 8.44 times 10 to the negative seven farads.
