Animated Solution for Physics - Electromagnetic Induction: An inductance coil has a reactance of $100\Omega$. When an AC signal of frequency $1000\text{Hz}$ is applied to the coil, the applied voltage leads the current by $45^{\circ }$. The self-inductance of the coil is

The problem of the "real" inductance coil is a classic trap in competitive physics. When a problem states that an "inductance coil has a reactance of $100\Omega$", it is tempting to immediately write $X_L=100\Omega$. However, a physical coil is made of wire, and wire has resistance. Therefore, a real coil is modeled as an ideal inductor $L$ in series with a resistor $R$.

The "reactance" mentioned in the problem statement is actually referring to the total opposition to the current, which is the impedance $Z$ of the coil. Thus, we must start with the premise that $Z=100\Omega$.

The problem gives us a crucial piece of information: the applied voltage leads the current by $45^{\circ }$. In an AC circuit containing both resistance and inductance, the phase angle $\varphi$ between the voltage and the current is given by the tangent relation:

Substituting $\varphi =45^{\circ }$, we get:

Since $\tan 45^{\circ }=1$, this beautifully simplifies our circuit model. It tells us that the inductive reactance is exactly equal to the resistance:

Now that we know $X_L=R$, we can return to our initial realization about the impedance. The impedance $Z$ of a series RL circuit is given by the Pythagorean sum of the resistance and the reactance:

We know $Z=100\Omega$. Substituting $R=X_L$ into the equation, we get:

Solving for $X_L$, we find the true inductive reactance of the coil:

With the true inductive reactance in hand, finding the self-inductance $L$ is a straightforward application of the reactance formula. We know that:

The frequency $f$ is given as $1000\text{Hz}$. Substituting our known values:

Now, we carefully isolate $L$:

Plugging in the standard approximations $\sqrt{2}\approx 1.414$ and $\pi \approx 3.14$:

This matches perfectly with option (a). The key takeaway here is to always read "coil" as a combination of $L$ and $R$ unless it is explicitly stated to be an "ideal" inductor!