what is the maximum value of charge on the capacitor given(at t=0,i=io,and capacitor is uncharged).the figure contains an inducter l and two capacitors c and 2c on either side of the inductor

To determine the maximum value of charge on the capacitor in the given circuit, we need to analyze the behavior of the LC circuit formed by the inductor and the capacitors. The scenario describes an inductor \( L \) with two capacitors, \( C \) and \( 2C \), positioned on either side. Initially, at \( t = 0 \), the current is \( i = i_0 \) and the capacitors are uncharged. Let's break this down step by step.

Understanding the Circuit Dynamics

In an LC circuit, energy oscillates between the inductor and the capacitors. When the circuit is energized, the current flows through the inductor, creating a magnetic field. As the current changes, this magnetic field induces a voltage that influences the capacitors.

Initial Conditions

At \( t = 0 \), since the capacitors are uncharged, the voltage across both capacitors is zero. The entire current \( i_0 \) flows through the inductor, and we can express the initial conditions mathematically:

• Voltage across \( C \): \( V_C(0) = 0 \)
• Voltage across \( 2C \): \( V_{2C}(0) = 0 \)

Energy Conservation in the Circuit

As the current flows, energy is transferred from the inductor to the capacitors. The total energy stored in the inductor at maximum current \( i_0 \) is given by:

Energy in the inductor: \( E_L = \frac{1}{2} L i_0^2 \)

This energy will eventually be shared between the two capacitors when the current reaches zero, at which point the maximum charge \( Q_{\text{max}} \) will be on the capacitors.

Charge Distribution on Capacitors

When the current is zero, the voltage across the capacitors can be expressed as:

• For capacitor \( C \): \( V_C = \frac{Q}{C} \)
• For capacitor \( 2C \): \( V_{2C} = \frac{Q_{2C}}{2C} \)

Since the capacitors are in series, the voltage across them must be equal when the current stops flowing. Therefore, we can set up the equation:

Voltage equality: \( V_C = V_{2C} \)

Substituting the expressions for voltage, we have:

\( \frac{Q}{C} = \frac{Q_{2C}}{2C} \)

From this, we can express \( Q_{2C} \) in terms of \( Q \):

\( Q_{2C} = 2Q \)

Total Charge Conservation

The total charge \( Q_{\text{total}} \) in the system must equal the initial charge that was stored in the inductor:

Total charge: \( Q + Q_{2C} = Q + 2Q = 3Q \)

Setting this equal to the maximum charge derived from the energy in the inductor gives us:

\( 3Q = \frac{1}{2} L i_0^2 \)

Calculating Maximum Charge

Now, we can solve for \( Q \):

\( Q = \frac{1}{6} L i_0^2 \)

Thus, the maximum charge on the capacitor \( C \) is:

Maximum charge on capacitor \( C \): \( Q_{\text{max}} = \frac{1}{6} L i_0^2 \)

And for capacitor \( 2C \), the maximum charge would be:

Maximum charge on capacitor \( 2C \): \( Q_{2C} = \frac{1}{3} L i_0^2 \)

Conclusion

In summary, the maximum charge on the capacitor \( C \) is \( \frac{1}{6} L i_0^2 \). This analysis illustrates how energy conservation and the characteristics of capacitors in series play a crucial role in determining the charge distribution in an LC circuit.