Question

A $20nF$capacitor is connected across an AC generator that produces a peak voltage of $5.0V$.

a. At what frequency $f$is the peak current $50mA$?

b. What is the instantaneous value of the emf at the instant when$i_c=I_c$?

Short answer

(a) The peak current $50mA$is at $f=7.95\times {10}^{4}Hz$.

(b) The instantaneous value of the emf at the instant ($i_c=I_c$)is $v_c=0$.

Step-by-step solution

Part(a) Step 1: Given information

We have been given that a $20nF$capacitor is connected across an AC generator that produces a peak voltage of $5.0V$.

We need to find at what frequency $f$is the peak current $50mA$.

Part(a) Step 2: Simplify

We know the formula :

$I_0=\omega V_{c}C=(2\pi f)V_{c}C$ (Let this equation be ($1$))

Readjusting equation ($1$), we get

$f=\frac{I_0}{2\pi V_{c}C}$ (Let this equation be ( $2$))

Substituting the values of localid="1649928656241" $I_0,V_c,C$in equation (localid="1649928665048" $2$), we get

localid="1649928671059" $f=\frac{50\times {10}^{-3}}{2\pi \left(5\right)(20\times {10}^{-9})}$

On simplifying,

localid="1649928677425" $f=7.95\times {10}^{4}Hz$

localid="1649928683686" $f=7950Hz$

Part(b) Step 1: Given information

We have been given that a $20nF$capacitor is connected across an AC generator that produces a peak voltage of $5.0V$.

We need to find the instantaneous value of the emf at the instant when$i_c=I_c$.

Part(b) Step 2: Simplify

We know that the current $i_c$can be expressed as :

$i_c=I_c\cos \left(\omega t-\frac{\pi }{2}\right)$ (Let this equation be ($3$))

Substituting $i_c=I_c$in equation ($3$), we get

$i_c=i_c\cos \left(\omega t-\frac{\pi }{2}\right)$

Dividing $i_c$on both sides, we get

$1=\cos \left(\omega t-\frac{\pi }{2}\right)$

Taking ${\cos }^{-1}$on both sides, we get

${\cos }^{-1}\left(1\right)=\omega t-\frac{\pi }{2}$

On simplifying,

$0=\omega t-\frac{\pi }{2}$

$\omega t=\frac{\pi }{2}$

It signifies that, $\omega t\to \frac{\pi }{2}$when $i_c=I_c$.

The instantaneous value of the emf is,

$v_c=V_c\cos \left(\omega t\right)$

localid="1649929903492" $v_c=V_c\cos \left(\frac{\pi }{2}\right)$

$v_c=0$