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### **Solution for Question 2(b):**

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#### **(i) Given:**
- \( q_1(t) = e^{-2t} \)
- \( q_2(t) = 10 \cos(2t) \)
- Find \( q(t) \) and \( i(t) \), where \( i(t) = \frac{dq(t)}{dt} \), for \( 0 \leq t < 30 \).

1. **Expression for Total Charge \( q(t) \):**  
   The total charge is given by:  
   \[
   q(t) = q_1(t) + q_2(t)
   \]  
   Substituting the given functions:  
   \[
   q(t) = e^{-2t} + 10 \cos(2t)
   \]

2. **Expression for Current \( i(t) \):**  
   The current is the time derivative of the charge:  
   \[
   i(t) = \frac{dq(t)}{dt}
   \]  
   Differentiating \( q(t) \):  
   \[
   i(t) = \frac{d}{dt} \left( e^{-2t} \right) + \frac{d}{dt} \left( 10 \cos(2t) \right)
   \]  
   Calculate each term:  
   \[
   \frac{d}{dt} \left( e^{-2t} \right) = -2e^{-2t}, \quad \frac{d}{dt} \left( 10 \cos(2t) \right) = -20 \sin(2t)
   \]  
   Therefore:  
   \[
   i(t) = -2e^{-2t} - 20 \sin(2t)
   \]

3. **Summary:**
   - \( q(t) = e^{-2t} + 10 \cos(2t) \)
   - \( i(t) = -2e^{-2t} - 20 \sin(2t) \)

#### **Sketch for \( 0 \leq t < 30 \):**
- \( q(t) \) combines an exponentially decaying term and a cosine wave.
- \( i(t) \) combines an exponentially decaying term and a sine wave.

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#### **(ii) Given:**
- Current \( i = 1.5 \, \mu A = 1.5 \times 10^{-6} \, \text{A} \),
- Time interval \( t = 2 \, \text{ns} = 2 \times 10^{-9} \, \text{s} \),
- Charge of an electron \( e = 1.6 \times 10^{-19} \, \text{C} \).

Find the number of electrons \( n \).

1. **Relation Between Current, Charge, and Time:**  
   The total charge \( Q \) is given by:  
   \[
   Q = i \cdot t
   \]  
   Substituting the given values:  
   \[
   Q = (1.5 \times 10^{-6}) \cdot (2 \times 10^{-9}) = 3 \times 10^{-15} \, \text{C}
   \]

2. **Number of Electrons \( n \):**  
   The number of electrons is:  
   \[
   n = \frac{Q}{e}
   \]  
   Substituting \( Q = 3 \times 10^{-15} \) and \( e = 1.6 \times 10^{-19} \):  
   \[
   n = \frac{3 \times 10^{-15}}{1.6 \times 10^{-19}} = 1.875 \times 10^4
   \]

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### **Final Answers:**
- (i)  
  - \( q(t) = e^{-2t} + 10 \cos(2t) \),  
  - \( i(t) = -2e^{-2t} - 20 \sin(2t) \).  
  - Sketch involves an exponential decay added to oscillatory components.

- (ii) Number of electrons \( n = 1.875 \times 10^4 \).

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Do you need detailed plots or further clarifications?  

Here are **related questions**:  
1. How do you calculate the derivative of more complex charge equations?  
2. Can we integrate \( i(t) \) to retrieve \( q(t) \) if \( q(t) \) is unknown?  
3. What happens to \( q(t) \) as \( t \to \infty \)?  
4. How does the frequency of oscillation affect \( i(t) \) and \( q(t) \)?  
5. What are the physical implications of negative values for \( i(t) \)?  

**Tip:** Always double-check units when working with nanoseconds, microamps, and Coulombs to avoid scaling errors!

Explore the solution for an electrical problem involving exponential and trigonometric charge functions and electron flow. This detailed explanation covers differentiation, current, charge relationships, and more. Perfect for engineering or physics students.

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