To solve the equation given by

$L \frac{dI}{dt} + RI = V,$

we'll follow these steps:

1. Rewrite the equation: This is a first-order linear ordinary differential equation (ODE). Rearranging it gives:

$\frac{dI}{dt} + \frac{R}{L}I = \frac{V}{L}.$

2. Identify the integrating factor: The integrating factor, $\mu(t)$, is given by:

$\mu(t) = e^{\int \frac{R}{L} dt} = e^{\frac{R}{L}t}.$

3. Multiply the ODE by the integrating factor:

$e^{\frac{R}{L}t} \frac{dI}{dt} + \frac{R}{L} e^{\frac{R}{L}t} I = \frac{V}{L} e^{\frac{R}{L}t}.$

4. Rewrite the left-hand side:

$\frac{d}{dt} \left( e^{\frac{R}{L}t} I \right) = \frac{V}{L} e^{\frac{R}{L}t}.$

5. Integrate both sides:

$e^{\frac{R}{L}t} I = \int \frac{V}{L} e^{\frac{R}{L}t} dt = \frac{V}{R} e^{\frac{R}{L}t} + C,$

where $C$ is a constant of integration.

6. Solve for $I(t)$:

$I(t) = \frac{V}{R} + Ce^{-\frac{R}{L}t}.$

7. Apply the initial condition $I(0) = 0$:

At $t = 0$:

$0 = \frac{V}{R} + C \implies C = -\frac{V}{R}.$

Thus, the expression for $I(t)$ becomes:

$I(t) = \frac{V}{R} - \frac{V}{R} e^{-\frac{R}{L}t} = \frac{V}{R} \left( 1 - e^{-\frac{R}{L}t} \right).$

Limit of $I(t)$

To find the limit as $t$ approaches infinity:

$\lim_{t \to \infty} I(t) = \frac{V}{R} \left( 1 - 0 \right) = \frac{V}{R}.$

Time to reach 50% of the limit value

To find when $I(t)$ reaches 50% of its limit value $\frac{V}{R}$:

$0.5 \cdot \frac{V}{R} = \frac{V}{R} \left( 1 - e^{-\frac{R}{L}t} \right).$

Dividing both sides by $\frac{V}{R}$ (assuming $V \neq 0$) gives:

$0.5 = 1 - e^{-\frac{R}{L}t}.$

Solving for $e^{-\frac{R}{L}t}$:

$e^{-\frac{R}{L}t} = 0.5 \implies -\frac{R}{L}t = \ln(0.5) \implies t = -\frac{L}{R} \ln(0.5).$

This simplifies to:

$t = \frac{L}{R} \ln(2).$

Summary

• The current $I(t)$ is given by:

$I(t) = \frac{V}{R} \left( 1 - e^{-\frac{R}{L}t} \right).$

• The limit value is $\frac{V}{R}$.
• The time to reach 50% of this limit is $t = \frac{L}{R} \ln(2)$.

Do you want more details or have any questions? Here are some related questions you might consider:

1. What physical principles are illustrated by the behavior of current in this circuit?
2. How does changing the resistance $R$ affect the time to reach 50% of the limit value?
3. What happens to the current if the voltage $V$ is increased?
4. How does the inductance $L$ influence the transient response of the circuit?
5. Can you explain the concept of time constant in the context of this circuit?

Tip: Understanding how inductors and resistors interact in circuits can provide insights into more complex electrical systems.