A solenoid of resistance `50 Omega` and inductance 80 H is connected to a 200 V battery, How long will it take for the current to reach 50% of its final equlibrium value ? Calculate the maximum enargy stored ?

Answer

To solve the problem step-by-step, we will first determine the time it takes for the current to reach 50% of its final equilibrium value, and then we will calculate the maximum energy stored in the solenoid.

### Step 1: Determine the Final Equilibrium Current
The final equilibrium current \( I_f \) in the circuit can be calculated using Ohm's law:
\[ I_f = \frac{V}{R} \]
where:
- \( V = 200 \, \text{V} \) (voltage of the battery)
- \( R = 50 \, \Omega \) (resistance of the solenoid)

Substituting the values:
\[ I_f = \frac{200}{50} = 4 \, \text{A} \]

### Step 2: Determine the Time Constant \( \tau \)
The time constant \( \tau \) for an RL circuit is given by:
\[ \tau = \frac{L}{R} \]
where:
- \( L = 80 \, \text{H} \) (inductance of the solenoid)
- \( R = 50 \, \Omega \) (resistance)

Substituting the values:
\[ \tau = \frac{80}{50} = 1.6 \, \text{s} \]

### Step 3: Calculate the Time to Reach 50% of Final Current
The current \( I(t) \) in an RL circuit as a function of time is given by:
\[ I(t) = I_f \left(1 - e^{-\frac{t}{\tau}}\right) \]
To find the time \( t \) when the current reaches 50% of its final value:
\[ I(t) = 0.5 I_f \]
Substituting \( I_f = 4 \, \text{A} \):
\[ 0.5 \times 4 = 4 \left(1 - e^{-\frac{t}{\tau}}\right) \]
This simplifies to:
\[ 2 = 4 \left(1 - e^{-\frac{t}{1.6}}\right) \]
Dividing both sides by 4:
\[ 0.5 = 1 - e^{-\frac{t}{1.6}} \]
Rearranging gives:
\[ e^{-\frac{t}{1.6}} = 0.5 \]
Taking the natural logarithm of both sides:
\[ -\frac{t}{1.6} = \ln(0.5) \]
Thus:
\[ t = -1.6 \ln(0.5) \]
Using \( \ln(0.5) \approx -0.693 \):
\[ t = -1.6 \times (-0.693) \approx 1.109 \, \text{s} \]

### Step 4: Calculate the Maximum Energy Stored
The maximum energy \( U \) stored in the inductor is given by:
\[ U = \frac{1}{2} L I_f^2 \]
Substituting the values:
\[ U = \frac{1}{2} \times 80 \times (4)^2 \]
Calculating:
\[ U = \frac{1}{2} \times 80 \times 16 = 640 \, \text{J} \]

### Final Answers
1. The time taken for the current to reach 50% of its final equilibrium value is approximately **1.11 seconds**.
2. The maximum energy stored in the solenoid is **640 Joules**.