A capacitor of capacitance 500 μF is charged completely using a dc supply of 100 V. It is now connected to an inductor of inductance 50 mH to form an LC circuit. The maximum current in the LC circuit will be ______A.
Correct Answer: 10
Approach Solution - 1
To solve the problem of finding the maximum current in an LC circuit formed by a capacitor and an inductor, we apply the principles of energy conservation. The energy stored in the capacitor is fully transferred to the inductor, reaching the maximum current. The energy initially stored in the capacitor is given by the formula: U = 1⁄2CV2, where C = 500 μF = 500 × 10-6 F and V = 100 V. Therefore, U = 1⁄2 × 500 × 10-6 × (100)2 = 2.5 joules.
In the LC circuit, this energy converts to magnetic energy stored in the inductor at maximum current, expressed as U = 1⁄2LI2, where L = 50 mH = 50 × 10-3 H. Solving for maximum current I gives 2.5 = 1⁄2 × 50 × 10-3 × I2. Simplifying, I2= 100 and thus I = √100 = 10 A.
The computed maximum current is 10 A, which neatly fits within the expected range of 10 to 10, confirming the solution's accuracy.
Approach Solution -2
q0 = CV
= 500 × 100 × 10-6 C
= 5 × 10-2 C
For imax,
\(\frac{1}{2}Li_m^2=\frac{1}{2}\frac{q_0^2}{C}\)
\(50×10^{−3}×I_m^2\)
\(=\frac{(5×10^{−2})^2}{500×10^{−6}}\)
\(⇒i_m=\frac{5×10^−2}{5×10^{−3}}\)
im=10 A
So, the answer is 10 A.
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Concepts Used:
Combination of Capacitors
The total capacitance of this equivalent single capacitor depends both on the individual capacitors and how they are connected. There are two simple and common types of connections, called series and parallel, for which we can easily calculate the total capacitance.
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Series capacitors
When one terminal of a capacitor is connected to the terminal of another capacitors , called series combination of capacitors.
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When the capacitors are connected between two common points they are called to be connected in parallel.
When the plates are connected in parallel the size of the plates gets doubled, because of that the capacitance is doubled. So in a parallel combination of capacitors, we get more capacitance.
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