Question

A solenoid has an inductance of 50 mH and a resistance of 0.025 Ω. If it is connected to a battery, how long will it take for the current to reach one half of its final equilibrium value

• A. 1.34 ms
• B. 1.2 ms
• C. 6.32 ms
• D. 0.23 ms

Answer 1

Answer: (A)

1.34 ms

Answer 2

$i = i_{0}\left( 1 - e^{- \frac{t}{\tau}} \right)$ where $i = \frac{1}{2}i_{0}$ and $\tau = \frac{L}{R}$. Thus

$\frac{1}{2}i_{0} = i_{0}\left( 1 - e^{- \frac{t}{\tau}} \right)$ or $\frac{1}{2} = e^{- \frac{t}{\tau}}$ or $2 = e^{+ \frac{t}{\tau}}$Thus

$t = \tau\log_{e}2 = \frac{50 \times 10^{- 3}}{0.025} \times 0.693 = 1.34 \times 10^{- 3}s = 1.34millisecond.$