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This is a problem related to a series RL circuit where a voltage of 90V is applied across a resistor \( R = 60 \, \Omega \) and an inductor \( L = 30 \, H \). We are asked to find the expression for the current through the inductor, \( i_L(t) \).

### Step-by-step analysis:

1. **Time Constant ( \( \tau \) )**:  
   For an RL circuit, the time constant \( \tau \) is given by:
   \[
   \tau = \frac{L}{R}
   \]
   Plugging in the values:
   \[
   \tau = \frac{30}{60} = 0.5 \, \text{seconds}
   \]

2. **Steady-State Current**:  
   The steady-state current \( I_{ss} \) in the circuit can be calculated using Ohm's law:
   \[
   I_{ss} = \frac{V}{R} = \frac{90}{60} = 1.5 \, \text{A}
   \]

3. **Current Expression**:  
   In a series RL circuit, the current \( i_L(t) \) as a function of time when a step voltage is applied is given by:
   \[
   i_L(t) = I_{ss} \left( 1 - e^{-\frac{t}{\tau}} \right)
   \]
   Substituting the known values of \( I_{ss} = 1.5 \, A \) and \( \tau = 0.5 \, s \):
   \[
   i_L(t) = 1.5 \left( 1 - e^{-2t} \right)
   \]

Thus, the correct expression for the current is:
\[
\boxed{1.5 \left( 1 - e^{-2t} \right) \, A}
\]

This matches option **a** in the given choices.

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Would you like more details or have any questions?

Here are 5 related questions to expand this topic:
1. How does the time constant \( \tau \) affect the response of an RL circuit?
2. What is the energy stored in the inductor at steady state?
3. How would the current change if the resistance \( R \) is increased?
4. How is the voltage across the inductor related to the current?
5. What happens to the circuit if the inductor is replaced by a capacitor?

**Tip:** In RL circuits, the current grows exponentially with time, reaching its maximum value as the inductor opposes changes in current initially but allows a steady flow after a long time.

A series RL circuit has a constant voltage V = 90V applied at t = 0, with a resistor R = 60 Ω and an inductor L = 30H. The expression of the current through the inductor is given by:

Learn how to find the current expression in a series RL circuit where a 90V voltage is applied at t=0. The circuit contains a 60Ω resistor and a 30H inductor. Step-by-step analysis with key formulas like time constant, steady-state current, and exponential growth/decay in RL circuits.

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