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Q3P

Expert-verifiedFound in: Page 936

Book edition
10th Edition

Author(s)
David Halliday

Pages
1328 pages

ISBN
9781118230718

**In a certain oscillating LC circuit, the total energy is converted from electrical energy in the capacitor to magnetic energy in the inductor in ${\mathbf{1}}{\mathbf{.}}{\mathbf{50}}{\mathbf{}}{\mathit{\mu}}{\mathit{s}}$**

- The period of oscillation is $6.00\mu s$.
- The frequency of the oscillation is $1.67\times {10}^{5}\text{Hz}$.
- The magnetic energy will be a maximum again at $3.00\mu s$ .

The total energy in conserved from electric energy in the capacitor to magnetic energy in the inductor in $t=1.50\mu s$.

**From the given condition, we can find the period of oscillation. Using the formula for the frequency of the oscillation, we can find the frequency of the oscillation, and using the relation between the energy stored in the magnetic field of the inductor at any time and the current through it, we can find the time after which the magnetic energy is a maximum again.**

Formulae:

The frequency of an oscillation in an LC circuit, ${\mathit{f}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{T}}$ (i)

The energy stored in the magnetic field of the inductor at any time, ${{\mathit{U}}}_{{\mathbf{B}}}{\mathbf{=}}\frac{\mathbf{L}{\mathbf{i}}^{\mathbf{2}}}{\mathbf{2}}$ (ii)

The total energy is conserved from electric energy in the capacitor to magnetic energy in the inductor $1.50\mu s$.

Thus, the period of oscillation is given by:

$T=4\left(1.50\mu s\right)\phantom{\rule{0ex}{0ex}}=6.00\mu s$

Hence, the value of the period of oscillation is localid="1662757013743" $6.00\mu s$.

The frequency of the oscillation is given using equation (i) as follows:

$f=\frac{1}{\left(6.00\times {10}^{-6}\text{s}\right)}\phantom{\rule{0ex}{0ex}}=1.67\times {10}^{5}\text{Hz}\phantom{\rule{0ex}{0ex}}$

Hence, the value of the frequency is $1.67\times {10}^{5}\text{Hz}$.

From equation (ii), we can get that ${U}_{B}\propto {i}^{2}$.

Thus, the magnetic energy does not depend on the direction of the current, so this will occur after one-half of a period. Therefore, the magnetic energy is a maximum again after the time:

$3.00\mu s$

Hence, the value of the time is $3.00\mu s$.

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