From the provided image, the problem seems to involve an RC circuit and is described by the following first-order differential equation:

$R \frac{dq}{dt} + \frac{1}{C} q = E(t)$

Data Given:

• $E(t) = 100 \, \text{V}$ (constant voltage)
• $R = 200 \, \Omega$
• $C = 10^{-4} \, \text{F}$
• Initial conditions: $q(0) = 0$

We are tasked to find:

1. $q(t)$: the charge on the capacitor as a function of time.
2. $i(t)$: the current as a function of time.

Step-by-Step Solution

1. The General Differential Equation

The governing equation is: $R \frac{dq}{dt} + \frac{1}{C} q = E(t)$

Substitute the given values: $200 \frac{dq}{dt} + 10^4 q = 100$

Simplify: $\frac{dq}{dt} + 50q = 0.5$

2. Solve the Differential Equation

This is a first-order linear differential equation of the form: $\frac{dq}{dt} + P(t) q = Q(t)$ where $P(t) = 50$ and $Q(t) = 0.5$.

Solution Formula

The general solution is: $q(t) = e^{-\int P(t) \, dt} \left[ \int e^{\int P(t) \, dt} Q(t) \, dt + C \right]$

Step 1: Find the Integrating Factor

$e^{\int P(t) \, dt} = e^{\int 50 \, dt} = e^{50t}$

Step 2: Solve for $q(t)$

Using the integrating factor: $q(t) = e^{-50t} \left[ \int e^{50t} \cdot 0.5 \, dt + C \right]$

Integrate: $\int e^{50t} \cdot 0.5 \, dt = 0.5 \int e^{50t} \, dt = \frac{0.5}{50} e^{50t} = 0.01 e^{50t}$

Thus: $q(t) = e^{-50t} \left[ 0.01 e^{50t} + C \right]$

Simplify: $q(t) = 0.01 + C e^{-50t}$

Step 3: Apply Initial Condition

At $t = 0$, $q(0) = 0$: $0 = 0.01 + C e^{-50(0)}$ $C = -0.01$

Therefore, the charge is: $q(t) = 0.01 - 0.01 e^{-50t}$

3. Find $i(t)$

The current is given by: $i(t) = \frac{dq}{dt}$

Differentiate $q(t)$: $q(t) = 0.01 - 0.01 e^{-50t}$ $\frac{dq}{dt} = -0.01 \cdot (-50) e^{-50t} = 0.5 e^{-50t}$

Thus: $i(t) = 0.5 e^{-50t}$

Final Answers:

1. Charge $q(t)$: $q(t) = 0.01 \left( 1 - e^{-50t} \right) \, \text{C}$

2. Current $i(t)$: $i(t) = 0.5 e^{-50t} \, \text{A}$

Do you need further clarifications or details?

Here are 5 related questions to extend your understanding:

1. How does the time constant ($\tau = RC$) affect the rate of charge and current decay?
2. How would the solution change if the voltage $E(t)$ were time-dependent, such as $E(t) = 100 \sin(t)$?
3. What is the maximum value of $q(t)$ for large $t$?
4. How does the value of $R$ influence the circuit's behavior?
5. Can you derive $i(t)$ directly without solving for $q(t)$ first?

Tip: In RC circuits, the time constant $\tau = RC$ plays a crucial role in determining how quickly the system reaches its steady-state behavior.