Find the energy sotred in the inductor...?

In an LC circuit, a 1 microfarad capacitor is charged to its maximum 34 microcoloumbs while the switch is open. The capacitor is connected in series with a 2 H inductor. The switch is closed at t = 0. Find the energy stored in the inductor at t = T/4, where T is the period of circuit oscillations.





The energy stored in an inductor is given by the equation E = 1/2 LI^2. I know L, so I have to find I. This can be done using the equation I = -Imax sin(omega t). To find Imax, I used the formula Imax = Qmax * omega. I also know omega = 1/sqrt(LC). So all that's left to find is t. This can be done using t = T/4. T = 1/f, where f = omega/2pi. Working this through, t becomes (pi * sqrt(LC))/2. When plugging in all the numbers and solving for everything, I come up with an answer that isn't right. Any help would be appreciated.

Find the energy sotred in the inductor...?
Assuming the circuit is lossless (not enough info is given to evaluate loss, anyhow), the energy stored in this system is constant, and equal to





E = ½ Q² / C = ½ (34E-6)² / 1E-6 = 578 µJ.





As noted above, this energy is fixed, provided loss is negligible. Of course, I’m not intending to say energy stored in capacitor is unchanging. What I mean is that total energy in the system is invariable, just as in a given mass-spring combination energy remains reasonably constant as well. In this latter case, energy continuously changes from potential to kinetic, and back again





In an LC circuit, likewise, energy is continuously changing; energy stored in the capacitor electric field is converted to magnetic field energy, but at any time the sum of energies, electric and magnetic, is constant. This affords a most convenient way to determine the energy stored in the magnetic field, by subtraction.





Therefore, we need only to find the electric field energy at a given time; magnetic energy is then given by the difference between total system energy and electric field energy. This indirect approach will prove to be much simpler than the reckless direct approach of finding current, in order to evaluate energy by means of E = ½ L I ².





In the present case, luckily enough, we’ve been spared a lot of work, for it can be stated at once, practically from inspection, that energy in magnetic field is just 578 µJ at t = T/4. It is widely known that current and voltage waveforms, in a purely reactive circuit as this, are out-of-phase by 90°. This is tantamount to say that voltage attains its maximum value at the same instant current is momentarily zero, and conversely; current will be maximum when voltage is at zero-crossing point. Again, the mechanical analog, likely more familiar, can afford some insight.





That electric energy is entirely converted to magnetic energy exactly after T/4 seconds have elapsed, can be demonstrated by a number of ways. By definition, ωT = 2π rad. At t = T/4, ωt = ωT/4 = 2π/4, or π/2 rad, which is equivalent to 90°. Alternatively, if after one period a full cycle of 360° is completed, at T/4 phase angle is to be 90°.

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