Question

At what frequency f do a $1.0\mu F$capacitor and a $1.0\mu H$inductor have the same reactance? What is the value of the reactance at this frequency?

Short answer

Frequency ; f $=159kHz$

Reactance ; $X=9.98\times {10}^{-4}\Omega$

Step-by-step solution

:Given Information

We have been given that

Capacitance of capacitor,$\left(C\right)=1.0\mu F={10}^{-6}F$

Inductance of inductor,$\left(\mathrm{L}\right)=1.0\mu H={10}^{-6}H$

We need to find out the frequency at which the reactance of inductor and capacitor is same and the reactance at that frequency.

Explanation 

Given , $C=1\mu F$and $L=1\mu H$

As we know Capacitive Reactance ;

$X_C=\frac{1}{2\pi fC}.....\left(1\right)$

where f is frequency ;

C is capacitance;

$X_C$is capacitive reactance

Also,

Inductive Reactance ; $X_L=2\pi fL$ $.......\left(2\right)$

where f is frequency ;

L is inductance

Since, both Reactance are same $i.e{X}_L=X_C$

$$\begin{gathered} \Rightarrow 2\pi fL=\frac{1}{2\pi fC} \\ \Rightarrow f=\frac{1}{2\pi \sqrt{CL}}=\frac{1}{2(3.14)\sqrt{(1\times {10}^{-6}F)(1\times {10}^{-6}H)}} \\ \Rightarrow f=159kHz \end{gathered}$$

Now reactance at frequency ,f

$$\begin{gathered} X_L=X_C=2\pi fL \\ =2(3.14)(159\times {10}^{3}Hz)(1\times {10}^{-6}H) \\ =9.98\times {10}^{-4}\Omega \end{gathered}$$