Question

Part A ConstantsI Poriodic Table Detemine he frequency of oscillation A 525 pP capacitor is charged to 185 V and then quickly connected to a 145 mll inductor Express your answer with the appropriate units Value Units Reauest Answer Submit Part B Detemine the peak value of the curent Express your answer with the appropriate units. Value Units Reauest Answer Submit Part C Determine the maximum energy stored in the magnetic Seld of the inductor Express your answer with the appropriate units. Units Value Us RequestAnse Submi

Answer 1

GIVEN:

1. C = 525 pF = 5.25 x 10^-10 F

2. L = 145 mH = 1.45 x 10^-4 H

3. Potential Difference applied across capacitor, V₀ = 185 V

SOLUTION:

The charge initially stored by the capacitor, q₀ is given by

$q_{o}=CV_{0}$

On putting the values, we get,

$q_{o}=(5.25\times10^{-10}\;F)(185\;V)=(9.71\times10^{-8}\;C)$

Part A

We know that the angular frequency of an LC circuit is given by,

$\omega = \frac{1}{\sqrt{LC}}$

Substituting the values of L and C, and solving, we get,

$\omega = \frac{1}{\sqrt{(1.45\times10^{-4}\; H)(5.25\times10^{-10} \;F)}}= 3.62\times10^{6} \; rad/s$

Now, frequency is given by,

$f=\frac{\omega }{2\pi }=\frac{3.62\times10^{6} \; rad/s}{2\pi\; rad} = 5.76\times10^{5}\; s^{-1} = 577\; KHz$

Hence, the frequency of oscillation is 577 KHz.

Part B

We know that the value of current, I is maximum when all the current is stored in the inductor.

From the conservation of energy we get,

$\frac{1}{2}LI_{o}^{2}=\frac{1}{2}\frac{q_{o}^{2}}{C}$

On simplification, we get,

$I_{o}=\frac{q_{o}}{\sqrt{LC}}=\omega q_{o}$

On putting the values, we get,

$I_{o}=(3.62\times10^{6} \; rad/s)(9.71\times10^{-8} \; C)=0.35\;A$

Hence, the peak current is 0.35 A

Part C

The energy stored in the magnetic field of the inductor will be maximum at the peak current.

$U_{max}=\frac{1}{2}LI_{o}^{2}$

Putting the values we get,

$U_{max}=\frac{1}{2}(1.45\times10^{-4}\; H)(0.35 A)^{2}=8.88\times10^{-6}\; J=8.88\;\mu J$

Hence, the maximum energy stored in the magnetic field of the inductor is 8.88µJ.