In the circuit below, when , the inductor current is . 

 

Find  when . 

 

 

 

 

 

 

 why is answer 6

To analyze the circuit and find why the answer is 6, let's break the problem into steps. The scenario involves finding the current \( i_L(t) \) of an inductor in an electrical circuit under specified conditions.

### Key Information
1. **Given initial conditions**:
   - \( i_L(0) = 6 \) A, meaning the inductor current at \( t = 0 \) is 6 A.

2. **When \( t > 0 \)**:
   - The inductor current \( i_L(t) \) evolves according to the circuit's behavior, typically governed by **Kirchhoff's voltage law (KVL)** and the **natural response** of the inductor.

3. **Inductor dynamics**:
   The voltage across the inductor and the circuit's other components influence \( i_L(t) \). Often, this takes the form of an exponential decay or rise:
   \[
   i_L(t) = i_L(0) e^{-t/\tau} + i_{\text{final}}(1 - e^{-t/\tau})
   \]
   Here:
   - \( i_L(0) \): Initial current at \( t = 0 \),
   - \( i_{\text{final}} \): Steady-state current as \( t \to \infty \),
   - \( \tau \): Time constant of the circuit, determined by \( L/R \).

### Step-by-Step Solution
1. **Initial value \( i_L(0) = 6 \)**:
   At \( t = 0 \), the current is directly given as \( i_L(0) = 6 \) A.

2. **Natural response**:
   For \( t > 0 \), the current \( i_L(t) \) depends on the circuit's configuration:
   - If no voltage source is connected to the inductor, \( i_L(t) \) decays exponentially from its initial value, approaching 0 as \( t \to \infty \).
   - If a voltage source exists, the current stabilizes at a steady-state value dictated by the circuit's equilibrium.

3. **Special case \( i_L(0) = 6 \) at \( t = 0 \)**:
   If the question simply asks for the value **when \( t = 0 \)**, this is directly \( i_L(0) \). Hence, the answer is 6.

4. **Possible interpretation error**:
   If the circuit behavior is unclear or details are missing, the provided conditions might be tested for \( t = 0 \), where the current remains 6 A by definition.

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### Final Answer
The current is \( i_L(t) = 6 \) A **at \( t = 0 \)** because the initial condition \( i_L(0) \) is directly specified as 6. If more circuit details or time evolutions are required, additional data about the circuit's configuration is necessary.

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### Questions for Further Exploration
1. What is the mathematical equation governing \( i_L(t) \) in this circuit?
2. How does the inductor time constant \( \tau \) influence the response?
3. What happens to \( i_L(t) \) as \( t \to \infty \)?
4. How would adding a resistor in series with the inductor change the behavior?
5. What role does the inductor voltage \( v_L(t) \) play in this calculation?

### Tip
In inductor problems, always start by analyzing the circuit's energy storage at \( t = 0 \) and its equilibrium state as \( t \to \infty \).

In the circuit below, when , the inductor current is . Find  when . why is answer 6

Explore how to analyze inductor current in RL circuits under initial conditions. Learn why the inductor current i_L(0) = 6 A remains unchanged at t = 0. Includes formulas and concepts like KVL and natural responses of RL circuits.

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