Answer:

Introduction
An oscillating LC (inductor-capacitor) circuit consists of an inductor and a capacitor connected in such a way that the charge on the capacitor varies over time, resulting in an oscillating current. In this circuit, energy is stored equally between the electric and magnetic fields.

Maximum Charge on the Capacitor
The maximum charge on the capacitor in an oscillating LC circuit is given by the formula:

Q = CV

Where Q is the charge on the capacitor, C is the capacitance of the capacitor, and V is the maximum voltage across the capacitor.

Energy Stored Equally Between Electric and Magnetic Fields
When the energy is stored equally between the electric and magnetic fields in an oscillating LC circuit, the energy stored in the capacitor is equal to the energy stored in the inductor. The energy stored in the capacitor is given by the formula:

E = (1/2)CV^2

Where E is the energy stored in the capacitor.

The energy stored in the inductor is given by the formula:

E = (1/2)LI^2

Where E is the energy stored in the inductor, L is the inductance of the inductor, and I is the maximum current in the circuit.

Since the energy stored in the capacitor is equal to the energy stored in the inductor, we can equate the two equations:

(1/2)CV^2 = (1/2)LI^2

Solving for Q, we get:

Q = √(L/C) * V

Therefore, the charge on the capacitor when the energy is stored equally between the electric and magnetic fields is:

Q = √(L/C) * V

Substituting Q = CV, we get:

V = Q/C

Therefore:

Q = √(L/C) * Q/C

Simplifying:

Q = Q/√(LC)

Therefore, the charge on the capacitor when the energy is stored equally between the electric and magnetic fields is:

Q/√(LC)

Since Q is given as the maximum charge on the capacitor, we can substitute Q = CV to get:

V = Q/C

Therefore:

Q/√(LC) = CV/√(LC)

Simplifying:

Q/√(LC) = V√(C/L)

Substituting V = Q/C, we get:

Q/√(LC) = Q/C * √(C/L)

Simplifying:

Q/√(LC) = Q/√(CL)

Therefore, the charge on the capacitor when the energy is stored equally between the electric and magnetic fields is:

Q/√(CL)

Since C = Q/V, we can substitute to get:

Q/√(LQ/V)

Simplifying:

Q/√(LV)

Therefore, the charge on the capacitor when the energy is stored equally between the electric and magnetic fields is:

Q/√(LV)

Since the frequency of oscillation in an LC circuit is given by:

f = 1/(2π√(LC))

We can substitute to get:

Q/√(4π^2f^2LV)

Simplifying:

Q/√(4π^2f^2L) * √(V