Lc Oscillations

Step 1: Recall the equation for mechanical oscillations of a block-spring system

The equation of motion for a simple harmonic oscillator (block-spring system) is given by:

where:

is the mass of the block
is the displacement from equilibrium position
is the force constant of the spring

Step 2: Recall the equation for LC oscillations in an electrical circuit

For an ideal LC circuit (ignoring resistance), the voltage across the capacitor plus the voltage across the inductor must be zero, based on Kirchhoff's Voltage Law.

Voltage across capacitor:

Voltage across inductor (induced emf):

Setting the sum of voltages to zero (ignoring sign conventions for now for analogy):

Step 3: Compare the equations for mechanical and LC oscillations

Comparing the mechanical oscillation equation:
and the LC oscillation equation:

We can observe the following analogies between the physical quantities:

Displacement ( ) in mechanical system is analogous to Charge ( ) in electrical system.
Mass ( ) in mechanical system is analogous to Inductance ( ) in electrical system.
Force constant ( ) in mechanical system is analogous to in electrical system.

Step 4: Identify the electrical equivalent of the force constant

From the analogy in Step 3, the electrical equivalent of the force constant ( ) of the spring is , which is the reciprocal of capacitance.

Step 5: Check the options against the derived equivalent

Reciprocal of capacitive reactance: Capacitive reactance . Reciprocal is . This is frequency dependent and not a constant equivalent to .

Capacitive reactance: . Frequency dependent, not a constant equivalent to .

Reciprocal of capacitance: . This matches our derived equivalent.

Capacitance: . This is analogous to the reciprocal of the force constant (if we consider stiffness as reciprocal of force constant).

Step 6: Select the correct option

Based on the analogy derived, the electrical equivalent of the force constant of the spring is the reciprocal of capacitance.

Final Answer: The final answer is reciprocal of capacitance

Step 1: Understand Energy Conservation in LC Oscillations

In an LC circuit without resistance, the total energy is constantly exchanged between the electric field energy stored in the capacitor and the magnetic field energy stored in the inductor. The total energy remains constant.

Step 2: Write the expressions for Energy stored in Capacitor and Inductor

Energy stored in the capacitor,

Energy stored in the inductor,

Total energy in the LC circuit,

Step 3: Use Initial Conditions to find Total Energy

At , charge on the capacitor , and current .

Total Energy

Step 4: Condition for Maximum Charge on Capacitor

The maximum charge on the capacitor occurs when the current in the inductor is zero. At this point, all the energy is stored in the capacitor in the form of electric field energy.

At maximum charge , current .

Total Energy

Step 5: Equate Total Energy and Solve for Maximum Charge

Since total energy is conserved:

Step 6: Choose the correct option

The calculated maximum charge is , which matches option (B).

Final Answer: The final answer is