e) If the maximum current $$ (100 \mathrm{~mA}) $$ is given by the formula $$ I=I_{ ext {max }} \sin \omega t $$, calculate the amount of current generated after; (i) 0.0025 seconds $$ = $$ (ii) 0.0125 seconds $$ = $$

Question

e) If the maximum current $$ (100 \mathrm{~mA}) $$ is given by the formula $$ I=I_{	ext {max }} \sin \omega t $$, calculate the amount of current generated after; (i) 0.0025 seconds $$ = $$ (ii) 0.0125 seconds $$ = $$

UpStudy AI · Accepted Answer

We are given that the current is defined by

$$
I(t) = I_{\text{max}} \sin (\omega t)
$$

with

$$
I_{\text{max}} = 100\,\text{mA}.
$$

Assuming that the circuit is powered by standard mains AC at $$50\,\text{Hz}$$, we have

$$
\omega = 2\pi f = 2\pi (50) = 100\pi\,\text{rad/s}.
$$

Thus, the current equation becomes

$$
I(t) = 100\,\text{mA} \cdot \sin(100\pi\, t).
$$

### (i) For $$t = 0.0025\,\text{s}$$

1. Substitute $$t = 0.0025$$ s into the argument of the sine:

$$
   100\pi \, t = 100\pi \times 0.0025 = 0.25\pi.
   $$

2. Evaluate the sine:

$$
   \sin(0.25\pi) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.7071.
   $$

3. Compute the current:

$$
   I(0.0025) = 100\,\text{mA} \times 0.7071 \approx 70.71\,\text{mA}.
   $$

### (ii) For $$t = 0.0125\,\text{s}$$

1. Substitute $$t = 0.0125$$ s into the argument:

$$
   100\pi \, t = 100\pi \times 0.0125 = 1.25\pi.
   $$

2. Recognize that

$$
   1.25\pi = \pi + 0.25\pi,
   $$

and use the identity

$$
   \sin(\pi + \theta) = -\sin \theta.
   $$

Therefore,

$$
   \sin(1.25\pi) = -\sin(0.25\pi) = -\frac{\sqrt{2}}{2} \approx -0.7071.
   $$

3. Compute the current:

$$
   I(0.0125) = 100\,\text{mA} \times (-0.7071) \approx -70.71\,\text{mA}.
   $$

### Final Answers

(i) For $$t = 0.0025$$ seconds, the generated current is approximately

$$
70.71\,\text{mA}.
$$

(ii) For $$t = 0.0125$$ seconds, the generated current is approximately

$$
-70.71\,\text{mA}.
$$
After 0.0025 seconds, the current is approximately 70.71 mA. After 0.0125 seconds, the current is approximately -70.71 mA.

e) If the maximum current \( (100 \mathrm{~mA}) \) is given by the formula \( I=I_{\text {max }} \sin \omega t \), calculate the amount of current generated after; (i) 0.0025 seconds \( = \) (ii) 0.0125 seconds \( = \)

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