An oscillating LC circuit is designed to operate at a peak current of 31 mA. The inductance of 42 mH is fixed and the frequency is varied by changing C. (a) If the capacitor has a maximum peak voltage of 50 V, we have to find whether the circuit can be safely operated at a frequency of 1.0 MHz. (b) We have to find the maximum safe operating frequency. (c) We have to determine the minimum capacitance.

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(a)

In a LC circuit the variation of charge with time on the capacitor is given by the function

where

The variation of the current in the circuit with time is given by the function

Therefore, the values of the maximum charge and the maximum current are related as

The circuit has been designed to operate at a peak current of 31 mA. Therefore, the maximum safe charge in the circuit at a frequency of 1.0 MHz will be

The inductance of 42 mH is fixed and the frequency is varied by changing C. The capacitance required for operating this system at frequency of 1 MHz can be found from the relation

The maximum voltage across the capacitor at this frequency will be

As the peak voltage rating of the capacitor is 50 V, the circuit cannot be safely operated at 1.0 MHz.

(b)

We will next find the maximum safe operating frequency . We will use the following relations in finding answer to this part of the problem.

We therefore have the relation

And

(c)

The minimum capacitance will therefore be