A solenoid has an inductance of 10 henty and a resistance of 2 ohm. It is connected to a 10 volt battery. How long will it take for the magnetic energy to reach `1//4` of its maximum value?

Explanation

To solve the problem, we need to find out how long it takes for the magnetic energy stored in the solenoid to reach \( \frac{1}{4} \) of its maximum value.

### Step-by-Step Solution:

1. **Understanding the Maximum Energy**:
The maximum energy \( U_0 \) stored in an inductor (or solenoid) is given by the formula:
\[ U_0 = \frac{1}{2} L I_0^2 \]
where \( L \) is the inductance and \( I_0 \) is the maximum current.

2. **Setting Up the Energy Condition**:
We want to find the time when the energy \( U \) is \( \frac{1}{4} U_0 \):
\[ U = \frac{1}{4} U_0 = \frac{1}{4} \left( \frac{1}{2} L I_0^2 \right) = \frac{1}{8} L I_0^2 \]

3. **Relating Energy to Current**:
The energy stored in the inductor at any time \( t \) is given by:
\[ U = \frac{1}{2} L I^2 \]
Setting this equal to \( \frac{1}{8} L I_0^2 \):
\[ \frac{1}{2} L I^2 = \frac{1}{8} L I_0^2 \]
Dividing both sides by \( \frac{1}{2} L \):
\[ I^2 = \frac{1}{4} I_0^2 \]
Taking the square root gives:
\[ I = \frac{I_0}{2} \]

4. **Current Decay in an RL Circuit**:
The current in an RL circuit as a function of time is given by:
\[ I(t) = I_0 \left(1 - e^{-\frac{R}{L}t}\right) \]
Setting \( I(t) = \frac{I_0}{2} \):
\[ \frac{I_0}{2} = I_0 \left(1 - e^{-\frac{R}{L}t}\right) \]
Dividing both sides by \( I_0 \) (assuming \( I_0 \neq 0 \)):
\[ \frac{1}{2} = 1 - e^{-\frac{R}{L}t} \]

5. **Solving for Time**:
Rearranging gives:
\[ e^{-\frac{R}{L}t} = \frac{1}{2} \]
Taking the natural logarithm of both sides:
\[ -\frac{R}{L}t = \ln\left(\frac{1}{2}\right) \]
Thus,
\[ t = -\frac{L}{R} \ln\left(\frac{1}{2}\right) \]

6. **Substituting Values**:
Given \( L = 10 \, \text{H} \) and \( R = 2 \, \Omega \):
\[ t = -\frac{10}{2} \ln\left(\frac{1}{2}\right) = -5 \ln\left(\frac{1}{2}\right) \]
We know that \( \ln\left(\frac{1}{2}\right) = -\ln(2) \):
\[ t = 5 \ln(2) \]
Using \( \ln(2) \approx 0.693 \):
\[ t \approx 5 \times 0.693 \approx 3.465 \, \text{seconds} \]

### Final Answer:
The time taken for the magnetic energy to reach \( \frac{1}{4} \) of its maximum value is approximately **3.465 seconds**.