Video Transcript
Which of the following formulas
correctly relates the impedance 𝑍 of a circuit to the circuit’s capacitive
reactance 𝑋 sub 𝐶, the circuit’s inductive reactance 𝑋 sub 𝐿, and the circuit’s
resistance 𝑅?
Each of our five answer options
purports to give the correct equation for the impedance of a circuit 𝑍. Impedance, we can recall, is an
overall measure of the opposition to the flow of charge in a circuit. It’s a term that applies for
alternating current circuits. And it accounts for, we could say
“sums up,” the total resistance and reactance of all components in the circuit.
Knowing that, we might think that
the impedance 𝑍 of a circuit simply equals the sum of 𝑋 sub 𝐶, 𝑋 sub 𝐿, and
𝑅. Note, however, that every single
one of our equations has a square root, that’s what this one-half power effectively
means, and that the terms inside the parentheses for each answer option are
squared.
We might be drawn then to answer
option (D), where we see that 𝑍 is claimed to be equal to the square root of 𝑅
squared plus 𝑋 𝐿 squared plus 𝑋 𝐶 squared. In fact, though, this is not the
correct equation for impedance. The correct equation for impedance
involves a subtraction. We take the capacitive reactance of
a circuit, and we subtract it from the inductive reactance. The correct equation for 𝑍 will
involve a term that looks like this
Notice this means that answer
option (B) is also eliminated. Here, we don’t subtract 𝑋 sub 𝐶
from 𝑋 sub 𝐿. In answer options (A), (C), and
(E), there is a subtraction. But in choice (A), this subtraction
is an 𝑋 sub 𝐿 term from an 𝑋 sub 𝐶 term. We know that order is reversed, so
(A) can’t be correct. And then in option (C), we do have
𝑋 sub 𝐶 being subtracted from 𝑋 sub 𝐿, but these terms are both squared. The correct equation for impedance
𝑍, though, has a term where 𝑋 sub 𝐶 to the first power, we could say, is
subtracted from 𝑋 sub 𝐿 to the first power.
The correct formula is given by
option (E), impedance 𝑍 equals the square root of 𝑅 squared plus the quantity 𝑋
sub 𝐿 minus 𝑋 sub 𝐶 squared.
