What is the instantaneous voltage across a 5-μF capacitor when the current through
it is i = 4 sin (106 t + 25◦) A?

To find the instantaneous voltage across the 5-μF capacitor given the current \( i(t) = 4 \sin (106 t + 25^\circ) \) A, we can use the relationship between current and voltage in a capacitor:

\[
i(t) = C \frac{d v(t)}{dt}
\]

where:
- \( i(t) \) is the current through the capacitor,
- \( C \) is the capacitance,
- \( v(t) \) is the voltage across the capacitor.

We can rearrange this equation to express voltage in terms of current:

\[
v(t) = \frac{1}{C} \int i(t) \, dt
\]

### Step 1: Substitute the given values
Given:
- Capacitance \( C = 5 \, \mu F = 5 \times 10^{-6} \, F \),
- \( i(t) = 4 \sin (106 t + 25^\circ) \).

We need to integrate the current to find the voltage:

\[
v(t) = \frac{1}{5 \times 10^{-6}} \int 4 \sin (106 t + 25^\circ) \, dt
\]

### Step 2: Integrate the current
The integral of \( \sin(\omega t + \phi) \) is:

\[
\int \sin (\omega t + \phi) \, dt = -\frac{1}{\omega} \cos (\omega t + \phi)
\]

Thus, integrating \( 4 \sin (106 t + 25^\circ) \):

\[
v(t) = \frac{1}{5 \times 10^{-6}} \left( -\frac{4}{106} \cos (106 t + 25^\circ) \right)
\]

### Step 3: Simplify the expression
Simplifying the constants:

\[
v(t) = \frac{-4}{5 \times 10^{-6} \times 106} \cos (106 t + 25^\circ)
\]

\[
v(t) = -\frac{4}{5.3 \times 10^{-4}} \cos (106 t + 25^\circ)
\]

\[
v(t) \approx -7547.17 \cos (106 t + 25^\circ) \, V
\]

### Final Answer:
The instantaneous voltage across the 5-μF capacitor is:

\[
v(t) \approx -7547.17 \cos (106 t + 25^\circ) \, V
\]

Would you like further details or clarification on any step?

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Here are 5 related questions to explore further:

1. What is the relationship between voltage and charge in a capacitor?
2. How does capacitance affect the phase difference between current and voltage in AC circuits?
3. What happens to the voltage across a capacitor when the frequency of the current increases?
4. How do you calculate the energy stored in a capacitor at any instant?
5. How does a capacitor behave in a DC versus an AC circuit?

**Tip:** In AC circuits, capacitors create a phase difference between current and voltage, with the current leading the voltage by 90°.

What is the instantaneous voltage across a 5-μF capacitor when the current through it is i = 4 sin (106 t + 25°) A?

Learn how to calculate the instantaneous voltage across a 5-μF capacitor when the current is given by i(t) = 4 sin(106t + 25°) A. This solution covers key concepts in AC circuits, capacitive reactance, and integration in calculus, suitable for undergraduate students.

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