The aim of this question is to develop an understanding of the **reactance of capacitors and inductors**. It also covers the concept of the **resonance frequency.**

The **reactance of an inductor** against the flow of alternating current can be computed using the **following formula**:

\[ X_{ L } \ = \ \omega \ L \]

The **reactance of a capacitor** against the flow of alternating current can be computed using the **following formula**:

\[ X_{ C } \ = \ \dfrac{ 1 }{ \omega \ C } \]

In above equations, $ X $ represents the **reactance**, $ \omega $ is the **frequency** in $ rad/sec $, $ L $ is the **inductance**, and $ C $ is the **capacitance**.

The **resonance frequency** is such a frequency where the **capacitive reactance** due to the capacitors and **inductive reactance** due to the inductance **becomes equal** in magnitude for a given circuit. Mathematically:

\[ X_{ L } \ = \ X_{ C } \]

## Expert Answer

**Part (a)** – The **reactance of an inductor** against the flow of alternating current can be computed using the **following formula**:

\[ X_{ L } \ = \ \omega \ L \]

**Since:**

\[ \omega \ =\ 2 \pi f \]

**So the above equation becomes:**

\[ X_{ L } \ = \ 2 \pi f \ L \]

**Given:**

\[ f \ = \ 60 \ Hz \]

\[ L \ = \ 0.45 \ H \]

**Substituting these values in the above equation:**

\[ X_{ L } \ = \ 2 \pi ( 60 ) \ ( 0.45 ) \]

\[ \Rightarrow X_{ L } \ = \ 169.65 \ \Omega \]

**Part (b)** – The **reactance of a capacitor** against the flow of alternating current can be computed using the **following formula**:

\[ X_{ C } \ = \ \dfrac{ 1 }{ \omega \ C } \]

**Since:**

\[ \omega \ =\ 2 \pi f \]

**So the above equation becomes:**

\[ X_{ C } \ = \ \dfrac{ 1 }{ 2 \pi f \ C } \]

**Given:**

\[ f \ = \ 60 \ Hz \]

\[ L \ = \ 2.5 \ \mu F \]

**Substituting these values in the above equation:**

\[ X_{ C } \ = \ \dfrac{ 1 }{ 2 \pi ( 60 ) \ ( 2.5 \mu ) } \]

\[ \Rightarrow X_{ C } \ = \ \dfrac{ 1 }{ 942.48 \ \mu } \]

\[ \Rightarrow X_{ C } \ = \ 1061.03 \ \Omega \]

## Numerical Results

\[ \Rightarrow X_{ L } \ = \ 169.65 \ \Omega \]

\[ \Rightarrow X_{ C } \ = \ 1061.03 \ \Omega \]

## Example

In the above question, find the **frequency where the reactance of both the inductor and capacitor becomes equal**.

**Given:**

\[ X_{ L } \ = \ X_{ C } \]

\[ 2 \pi f \ L \ = \ \dfrac{ 1 }{ 2 \pi f \ C } \]

\[ f^{ 2 } \ = \ \dfrac{ 1 }{ 4 \pi^{ 2 } \ L \ C } \]

\[ f \ = \ \dfrac{ 1 }{ 2 \pi \ \sqrt{ L \ C } } \]

**Substituting values:**

\[ f \ = \ \dfrac{ 1 }{ 2 \pi \ \sqrt{ ( 0.450 ) \ ( 2.5 \ \mu ) } } \]

\[ f \ = \ \dfrac{ 1 }{ 2 \pi \ ( 1.06 \ mili ) } \]

\[ f \ = \ \dfrac{ 1 }{ 6.664 \ mili ) } \]

\[ f \ = \ 150 \ Hz \]