Q25E A聽35V聽battery with negligible ... [FREE SOLUTION]

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University Physics with Modern Physics

Hugh D. Young, Roger A. Freedman

Physics

14th edition Edition 路 1596 pages

ISBN: 9780321973610

Chapter 4: Q25E (page 1014)

A $35 V$ battery with negligible internal resistance, a $50 惟$ resistor, and a $1.25 m H$ inductor with negligible resistance is all connected in series with an open switch. The switch is suddenly closed. (a) How long after closing the switch will the current through the inductor reach one-half of its maximum value? (b) How long after closing the switch will the energy stored in the inductor reach one-half of its maximum value?

Short Answer

Expert verified

a) $t = 1.73 脳 10^{- 5} s$

b) $t = 3.07 脳 10^{- 5} s$

Step by step solution

Calculate the time for one half current

In an LR circuit, the current increases by $i = \frac{蔚}{R} 1 - e^{- \frac{t}{蟿}}$ , where $蟿 = \frac{L}{R}$ .

Given that $L = 1.25 m H, R = 50 惟$ and $蔚 = 35 V$ .

For the current to be one half of the maximum value, $i = \frac{i_{m a x}}{2}$ .

$\frac{1}{2} 脳 \frac{蔚}{R} = \frac{蔚}{R} (1 - e^{- \frac{t}{T}}) \frac{1}{2} = 1 - e^{- \frac{t}{T}} e^{- \frac{t}{T}} = \frac{1}{2}$

Take the natural logarithm on both the sides.

$\frac{- t}{蟿} = \ln \frac{1}{2} t = \frac{L \ln 2}{R} t = \frac{1.25 脳 10^{- 3} 脳 \ln 2}{50} t = 1.73 脳 10^{- 5} s$

Calculate the time for half the maximum energy

For the energy to be one half of the maximum energy,

$U = \frac{U_{m a x}}{2} \frac{1}{2} L i^{2} = \frac{1}{2} L {i^{2}}_{m a x}^{} i = \frac{i_{m a x}}{\sqrt{2}}$

Now, from the relation of current and maximum current,

role="math" localid="1664252563069" $1 - e^{- \frac{t}{T}} = \frac{1}{\sqrt{2}} e^{- \frac{t}{T}} = 1 - \frac{1}{\sqrt{2}} e^{- \frac{t}{T}} = 0.2929$

Taking the natural logarithm on both the sides,

$t = \frac{L 脳 \ln 0.2929}{R} t = \frac{1.25 脳 10^{- 3} 脳 \ln 0.2929}{50} t = 3.07 脳 10^{- 5} s$

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Short Answer

Step by step solution

Calculate the time for one half current

Calculate the time for half the maximum energy

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