The time taken for the magnetic energy to reach 25% of its maximum value, when a solenoid of resistance R, inductance L is connected to a battery, is :

Solution

Physics sir · 4 steps

Energy goes as i², so 25% of the energy means only half the current. Find when i reaches I₀/2.

1. Energy to current. The energy stored in the solenoid is U = 1/2Li², so it is largest when the current is at its maximum I₀ = E/R. For U = 1/4Uₘₐₓ we need i² = 1/4I₀², that is i = I₀/2.
2. Growth of current. In an LR circuit connected to a battery, the current grows from zero as shown below, with time constant τ = L/R.

   i = I₀(1 - e^(-Rt/L))

3. Set i to half. Put i = I₀/2: 1 - e^(-Rt/L) = 1/2, so e^(-Rt/L) = 1/2. Taking logs, Rt/L = ln 2.
4. So. t = L/R ln 2, which is option (c).

The idea to keep. Whenever energy depends on the square of a growing quantity, convert the energy fraction to that quantity first: 25% energy means 50% current, 1% energy means 10% current.

Answer from the NTA final answer key. Solution by PrepWiser.