To solve the problem, we will follow these steps:
### Step 1: Identify the given values
- Inductance \( L = 10 \, \text{H} \)
- Resistance \( R = 2 \, \Omega \)
- Voltage \( V = 10 \, \text{V} \)
### Step 2: Calculate the maximum current \( I_0 \)
The maximum current \( I_0 \) can be calculated using Ohm's law:
\[
I_0 = \frac{V}{R} = \frac{10 \, \text{V}}{2 \, \Omega} = 5 \, \text{A}
\]
### Step 3: Calculate the maximum energy \( E_0 \)
The maximum energy stored in the inductor is given by the formula:
\[
E_0 = \frac{1}{2} L I_0^2
\]
Substituting the values:
\[
E_0 = \frac{1}{2} \times 10 \, \text{H} \times (5 \, \text{A})^2 = \frac{1}{2} \times 10 \times 25 = 125 \, \text{J}
\]
### Step 4: Calculate the energy at \( \frac{1}{4} \) of maximum value
To find the energy when it reaches \( \frac{1}{4} \) of its maximum value:
\[
E = \frac{E_0}{4} = \frac{125 \, \text{J}}{4} = 31.25 \, \text{J}
\]
### Step 5: Relate energy to current
The energy stored in the inductor can also be expressed as:
\[
E = \frac{1}{2} L I^2
\]
Setting this equal to \( 31.25 \, \text{J} \):
\[
31.25 = \frac{1}{2} \times 10 \times I^2
\]
Solving for \( I^2 \):
\[
I^2 = \frac{31.25 \times 2}{10} = 6.25 \quad \Rightarrow \quad I = \sqrt{6.25} = 2.5 \, \text{A}
\]
### Step 6: Use the formula for current growth 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 \( I(t) = 2.5 \, \text{A} \):
\[
2.5 = 5 \left(1 - e^{-\frac{R}{L}t}\right)
\]
Dividing both sides by 5:
\[
0.5 = 1 - e^{-\frac{R}{L}t}
\]
Rearranging gives:
\[
e^{-\frac{R}{L}t} = 0.5
\]
### Step 7: Take the natural logarithm
Taking the natural logarithm of both sides:
\[
-\frac{R}{L}t = \ln(0.5)
\]
Substituting \( R = 2 \, \Omega \) and \( L = 10 \, \text{H} \):
\[
-\frac{2}{10}t = \ln(0.5)
\]
\[
-\frac{1}{5}t = -0.693 \quad \Rightarrow \quad t = 0.693 \times 5 = 3.465 \, \text{s}
\]
### Step 8: Round to appropriate significant figures
Rounding gives approximately:
\[
t \approx 3.5 \, \text{s}
\]
### Final Answer:
The time taken for the magnetic energy to reach \( \frac{1}{4} \) of its maximum value is approximately **3.5 seconds**.
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