In an oscillating LC circuit in which , the maximum potential difference across the capacitor during the oscillations is and the maximum current through the inductor is . (a)What is the inductance L? (b)What is the frequency of the oscillations? (c) How much time is required for the charge on the capacitor to rise from zero to its maximum value?
In an oscillating LC circuit, energy is shuttled periodically between the electric field of the capacitor and the magnetic field of the inductor; instantaneous values of the two forms of energy are given by equations 31-1 and 31-2. By solving these equations and by substituting the values, we can calculate the inductance. From the relation between the linear frequency and the angular frequency, we can find the frequency. The period is reciprocal to frequency. From that, we can find the time required for the charge on the capacitor to rise from zero to its maximum value.
The magnetic energy stored in the inductor, (i)
The electric energy stored in the capacitor, (ii)
The frequency of the oscillation in the circuit, (iii)
The charge across the capacitor, (iv)
The angular frequency of the LC oscillations, (v)
The period of an oscillation, (vi)
Equating equations (i) and (ii), we can get the value of the inductance as follows:
Hence, the value of the inductance is .
Substituting value of equation (v) in equation (iii), we can get the frequency of the oscillations as follows:
Hence, the value of the frequency is .
From figure 31-1, we see that the required time is one fourth of a period, so the period of oscillation can be calculated using equation (vi) as follows:
Now, the required time to charge the capacitor to the maximum value as follows:
Hence, the value of the required time is .
An ac generator with emf amplitude and operating at frequency causes oscillations in a series circuit having , , and . Find (a) the capacitive reactance , (b) the impedance Z, and (c) the current amplitude I. A second capacitor of the same capacitance is then connected in series with the other components. Determine whether the values of (d) , (e) Z, and (f) I increase, decrease, or remain the same.
A series RLC circuit is driven by an alternating source at a frequency of 400Hz and an emf amplitude of 90.0V. The resistance is , the capacitance is , and the inductance is . What is the RMS potential difference across (a) the resistor, (b) the capacitor, and (c) the inductor? (d) What is the average rate at which energy is dissipated?
In Fig. 31-38, a three-phase generator G produces electrical power that is transmitted by means of three wires. The electric potentials (each relative to a common reference level) are for wire 1, for wire 2, and for wire 3. Some types of industrial equipment (for example, motors) have three terminals and are designed to be connected directly to these three wires. To use a more conventional two-terminal device (for example, a lightbulb), one connects it to any two of the three wires. Show that the potential difference between any two of the wires (a) oscillates sinusoidally with angular frequency and (b) has an amplitude of .
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