Question

A coil of inductance 0.5 h and a resistor of resistance 100ω are connected in series to a 240v,50 hz supply. (a) find the maximum current in the circuit (b) what is the time lag between voltage maximum and current maximum?

Answer 1

Step 1: Given

Inductance, L = 0.5H

Resistance, $R=100\Omega$

RMS voltage,$V_{rms}=240v$

Frequency, f = 50Hz

Step 2: Calculate impedance for LR circuit

angular frequency, $\omega =2\pi f$

Substitute values-

$$\begin{gathered} \omega =2\pi \times 50 \\ \omega =100\pi rad/s \end{gathered}$$

Inductive reactance, $X_L=\omega L$

Substitute values-

$$\begin{gathered} X_L=100\pi \times 0.5 \\ {X}_L=50\pi \end{gathered}$$

Now, the impedance for LR circuit, $z=\sqrt{X_L^2+R^2}$

Substitute values-

$$\begin{gathered} z=\sqrt{{\left(50\pi \right)}^2+{100}^2} \\ z=186.142\Omega \end{gathered}$$
Step 3: Calculate the Maximum Current In The Circuit

According to ohms law for AC, $V_{rms}=z.I_{rms}$, here $I_{rms}$ is the RMS current.

Substitute values-

$$\begin{gathered} 240=186.142\times I_{rms} \\ {I}_{rms}=1.289A \end{gathered}$$

Also, $$\begin{gathered} I_{rms}=\frac{I_{max}}{\sqrt{2}} \\ {I}_{max}=I_{rms}\sqrt{2} \end{gathered}$$

here $I_{max}$ is the maximum current.

Substitute values-

$$\begin{gathered} I_{max}=1.289\sqrt{2} \\ {I}_{max}=1.289\times 1.414 \\ {I}_{max}=1.8226A \end{gathered}$$

thus, the The Maximum Current In The Circuit is $1.8226A$.

Step 4: Calculate the Time Lag Between Voltage Maximum And Current Maximum

$$\begin{gathered} \cos \varphi =\frac{R}{z} \\ \cos \varphi =\frac{100}{186.142} \\ \varphi ={\cos }^{-1}\left(\frac{100}{186.142}\right) \\ \varphi =1radian \end{gathered}$$

now, the time lag, $$\begin{gathered} t=\frac{\varphi }{\omega } \\ t=\left(\frac{1}{100\pi }\right)s \end{gathered}$$.

Thus, the time lag is $\left(\frac{1}{100\pi }\right)s$.

Hence,

(A) The Maximum Current In The Circuit is $1.8226A$.

(B) The Time Lag Between Voltage Maximum And Current Maximum is $\left(\frac{1}{100\pi }\right)s$.