An inductor is hooked up to an ac voltage source. the voltage source has emf v0 and frequency f. the current amplitude in the inductor is i0.

Question

asked Sep 22, 2024 134k views

1 Answer

Andrew Lorien · Answer 1 · 2024-09-27T13:53:31+0000

The expressions for the reactance
$\left(X_L\right)$ and the inductance
$(L)$ in terms of
V_0$, $f , and
$I_0$ are:

1.
$X_L=(V_0)/(I_0)$

2.
$L=(V_0)/(2 \pi f I_0)$

The reactance of an inductor
$X_L$ in an AC circuit is given by the formula:

X_L=(V_0)/(I_0)

This formula relates the voltage across an inductor
$\left(V_0\right)$ to the current flowing through it
$\left(I_0\right)$ in an AC circuit.

The inductive reactance
$\left(X_L\right)$ is also related to the inductance
$(L)$ and frequency
$(f)$ in the following way:

$$X_L=2 \pi f L$

Now, we can use these formulas to derive the expressions for
$X_L$ and
$L$ in terms of
$V_0, f$ , and
$I_0$ :

From the first equation
$X_L=(V_0)/(I_0)$ , we can rearrange to solve for
$V_0$ :

X_L * I_0=V_0

So, the expression for
$X_L$ in terms of
$V_0$ and
$I_0$ is:

$X_L=(V_0)/(I_0)$

From the second equation
$X_L=2 \pi f L$ , we can rearrange to solve for
$L$ :

$L=(X_L)/(2 \pi f)$

Substitute the expression for
$X_L=(V_0)/(I_0)$ :

$L= (V_0)/(2 \pi f I_0)$

Substituting
$X_L=(V_0)/(I_0)$ into the expression for inductance
$L=(V_0)/(2 \pi f I_0)$ :

$L=(V_0)/(2 \pi f \cdot (V_0)/(I_0))$

Now, let's simplify this expression:

$L=(V_0 \cdot I_0)/(2 \pi f \cdot V_0)$

$V_0$ cancels out in the numerator and denominator:

$L=(I_0)/(2 \pi f)$

Therefore, after substitution, the expression for the inductance
$(L)$ in terms of
$V_0, f$ , and
$I_0$ simplifies to:

$L=(I_0)/(2 \pi f)$

The complete question is given below:

An inductor is hooked up to an AC voltage source. The voltage source has EMF
$V_0$ and frequency
. The current amplitude in the inductor is
I_0 .

Part A

What is the reactance
$X_L$ of the inductor?

Express your answer in terms of V0 and
I_0 .

Part B

What is the inductance
of the inductor?

Express your answer in terms of
$V_0$ ,
, and
I_0 .