Question

An electric motor has an effective resistance of $\mathbf{32}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\text{Ω}}$and an inductive reactance of$\mathbf{45}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\text{Ω}}$when working under load. The rms voltage across the alternating source is$\mathbf{420}\mathbf{}\mathbf{}\mathit{V}\mathbf{.}$Calculate the rms current.

Short answer

The rms current value is $7.61\text{A}$.

Step-by-step solution

The given data

1. Effective resistance $R=32.0\text{Ω}$
2. Inductive reactance$X_L=45.0\text{Ω}$
3. RMS value of voltage${\epsilon }_{rms}=420\text{V}$

Understanding the concept of RMS value

The RMS value of current means the square root of mean of square of value of current. The resistance offered in the path of current by the inductor is known as inductive reactance.

The rms current of a circuit,

$I_{rms}=\frac{{\epsilon }_{rms}}{Z}$ ...(i)

The impedance of the$LR$circuit for the driving frequency${\omega }_d$

$Z=\sqrt{R^2+{X_L}^2}$ ...(ii)

Calculation of the rms current

The impedance value of the circuit can be given using equation (ii) as follows:

$Z=\sqrt{R^2+{X_L}^2}$

Now, substituting the above value in equation (i), we can get the value of rms current for the given data as follows:

$$\begin{gathered} I_{rms}=\frac{{\epsilon }_{rms}}{\sqrt{R^2+{\left(X_L\right)}^2}} \\ =\frac{420\text{V}}{\sqrt{{\left(45\text{Ω}\right)}^2+{\left(32\text{Ω}\right)}^2}} \\ =7.61\text{A} \end{gathered}$$

Hence, the value of the current is localid="1662764908957" $7.61\text{A}$.