Question

The time taken for the magnetic energy to reach 25% of its maximum value in an LR circuit is:

• A. \(\frac{L}{R} \ln 2\)
• B. \(\frac{L}{R} \ln 5\)
• C. \(\frac{L}{R} \ln 10\)
• D. infinite

Hint:

Because energy depends on the {square} of the current, reaching 25% energy only requires the current to reach 50% of its maximum value.

Answer 1

The Correct Option is A

Step 1: Magnetic energy \(U = \frac{1}{2}LI^2\). Maximum energy \(U_0 = \frac{1}{2}LI_0^2\).
Step 2: Given \(U = 0.25 U_0 \implies \frac{1}{2}LI^2 = 0.25 \left(\frac{1}{2}LI_0^2\right) \implies I^2 = \frac{1}{4}I_0^2 \implies I = \frac{1}{2}I_0\).
Step 3: The growth of current in an LR circuit is \(I = I_0(1 - e^{-Rt/L})\).
Step 4: Substitute \(I = I_0/2\): \[\frac{1}{2}I_0 = I_0(1 - e^{-Rt/L}) \implies \frac{1}{2} = 1 - e^{-Rt/L} \implies e^{-Rt/L} = \frac{1}{2}\] \[\frac{Rt}{L} = \ln 2 \implies t = \frac{L}{R} \ln 2\]