An alternating emf is represented by e=420sin(942,478t) volts. Calculate

Question

a)frequency of the emf
b) periodic time
c) instantaneous value of the emf 1,2ms after commencement of a cycle

Answer 1

Given the alternating emf \( e(t) = 420 \sin(942,478t) \) volts, let's extract the required parameters:

### Part (a): Calculate the Frequency
The standard form of an alternating emf is given as:
\[ e(t) = E_0 \sin(2\pi ft) \]

By comparing the given equation with the standard form, we notice that:
\[ 942,478t = 2\pi ft \]

Thus, the angular frequency (\(\omega\)) is:
\[ \omega = 2\pi f \]
\[ 942,478 = 2\pi f \]

Solving for \(f\):
\[ f = \frac{942,478}{2\pi} \]
\[ f \approx \frac{942,478}{6.2832} \]
\[ f \approx 150,000 \text{ Hz} \]

So, the frequency of the emf is \( \boxed{150,000 \text{ Hz}} \).

### Part (b): Calculate the Periodic Time (\(T\))
The periodic time \( T \) is the reciprocal of the frequency:
\[ T = \frac{1}{f} \]
\[ T = \frac{1}{150,000} \text{ s} \]
\[ T \approx 6.6667 \times 10^{-6} \text{ s} \]
\[ T \approx 6.67 \text{ µs} \]

So, the periodic time is \( \boxed{6.67 \text{ µs}} \).

### Part (c): Calculate the Instantaneous Value of the emf 1.2 ms After Commencement of a Cycle
The instantaneous value of the emf is given by the expression:
\[ e(t) = 420 \sin(942,478t) \]

We'll substitute \( t = 1.2 \text{ ms} = 1.2 \times 10^{-3} \text{ s} \):
\[ e(t) = 420 \sin(942,478 \times 1.2 \times 10^{-3}) \]
\[ e(t) = 420 \sin(1130.9736) \]

To find the sine value, we typically convert the angle to a value within the range of \(0\) to \(2\pi \):
\[ 1130.9736 \, \text{radians} \]

Since \( 2\pi \approx 6.2832 \),
\[ \text{Reduced angle} = 1130.9736 \mod 2\pi \]
\[ \text{Reduced angle} = 1130.9736 \mod 6.2832 \]
\[ \text{Reduced angle} \approx 4.362 \, \text{radians} \]

Thus:
\[ e(t) = 420 \sin(4.362) \]

Using a calculator:
\[ \sin(4.362) \approx -0.944 \]
\[ e(t) \approx 420 \times -0.944 \]
\[ e(t) \approx -396.48 \text{ } \text{volts} \]

So, the instantaneous value of the emf 1.2 ms after commencement of a cycle is \( \boxed{-396.48 \text{ volts}} \).