Answer to Question #291938 in Electricity and Magnetism for araju

Question

A 50 Hz sinusoidal voltage has rms value of 200 V. At t=0, the instantaneous value is positive and half of its maximum value. Write down expression for voltage and sketch the waveform

Answer 1

Using the provided data, we proceed to find the elements for $V_{(t)}=V_{max}\sin{(\omega t+\phi)}$:

$\\ f = 50\, Hz \\ \therefore \omega=2\pi f= 314.16\,{ rad/s}$

We find the maximum value with V_rms and the phase angle as:

$V_{rms}=200\,V \\V_{max}=\sqrt{2}V_{rms}= (1.4142)(200\,V)=282.84\,V \\ \phi= \arcsin{\Big( \dfrac {V_{(0)}}{V_{max}}\Big)} \\ \text{} \\ \phi= \arcsin{\Big( \dfrac {½V_{max}}{V_{max}}\Big)} =\arcsin{(½)}= 30°=\frac{\pi}{6}$

In conclusion, the expression for the voltage will be $V_{(t)}=(282.84\,V)\sin{(100\pi t+\frac{\pi}{6})}$.

The graphic for the voltage will be: