In an oscillating LC circuit, the maximum charge on the capacitor is Q. The charge on the capacitor, when the energy is stored equally between the electric and magnetic field isa)Q/2b)Q/√2c)Q/√3d)Q/3Correct answer is option 'B'. Can you explain this answer?

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Answer

Answer

Answer

Oscillating LC Circuit:
An oscillating LC circuit consists of an inductor (L) and a capacitor (C) connected in parallel. When the circuit is initially charged with a certain amount of charge Q, the energy is stored in the electric and magnetic fields of the system.

Energy Storage in an LC Circuit:
The energy stored in an LC circuit is given by the equation:
E = (1/2) * Q^2 / C

Equal Energy Split:
When the energy is equally split between the electric and magnetic fields, it means that both fields store the same amount of energy. Let's assume that at a certain instant, the charge on the capacitor is q and the current through the inductor is i.

Energy in Electric Field:
The energy stored in the electric field of the capacitor is given by the equation:
E_electric = (1/2) * q^2 / C

Energy in Magnetic Field:
The energy stored in the magnetic field of the inductor is given by the equation:
E_magnetic = (1/2) * L * i^2

Equating Energies:
To find the charge on the capacitor when the energy is equally split, we can equate the two expressions for energy:

(1/2) * q^2 / C = (1/2) * L * i^2

Simplifying the equation gives:
q^2 / C = L * i^2

Since q = Cv (where v is the voltage across the capacitor) and i = L(di/dt) (where di/dt is the rate of change of current), we can substitute these expressions into the equation:

(Cv)^2 / C = L * (L(di/dt))^2

Simplifying further gives:
v^2 = L / C * (di/dt)^2

Charge on the Capacitor:
Since v = Q/C, we can substitute this expression into the equation:

(Q/C)^2 = L / C * (di/dt)^2

Simplifying gives:
Q^2 = L / C * C * (di/dt)^2

Cancelling out the C term gives:
Q^2 = L * (di/dt)^2

Taking the square root of both sides gives:
Q = √(L * (di/dt)^2)

Conclusion:
From the above equation, we can see that the charge on the capacitor (Q) is proportional to the square root of the energy stored in the magnetic field (L * (di/dt)^2). Therefore, when the energy is equally split between the electric and magnetic fields, the charge on the capacitor is Q/√2. Hence, the correct answer is option 'B'.

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