1. Circuit Analysis

The circuit contains 7 identical coils (inductance $L$) and one capacitor ($C$). Initially, the switch is in position 1, allowing currents $I_1$ and $I_2$ to flow in the coils (as setup by the batteries).

When the switch is thrown to position 2, the batteries are disconnected, and all 7 inductors are connected in parallel with the capacitor.

2. Equivalent Inductance & Current

Since all 7 inductors are identical and connected in parallel, the equivalent inductance is:

$$ L_{eq} = \frac{L}{7} $$

The total initial current in the system (the sum of currents flowing into the node) is $I_{total} = I_1 + I_2$. This total current will oscillate between the effective inductance and the capacitor.

3. Maximum Charge Calculation

We use the conservation of energy. The initial magnetic energy stored in the equivalent inductance system converts entirely into electrostatic energy in the capacitor at the moment of maximum charge.

$$ U_{magnetic} = U_{electric} $$ $$ \frac{1}{2} L_{eq} (I_{total})^2 = \frac{1}{2} \frac{Q_{max}^2}{C} $$ $$ \frac{1}{2} \left(\frac{L}{7}\right) (I_1 + I_2)^2 = \frac{Q_{max}^2}{2C} $$ $$ Q_{max}^2 = \frac{LC}{7}(I_1 + I_2)^2 $$ $$ Q_{max} = (I_1 + I_2)\sqrt{\frac{LC}{7}} $$

4. Current in the Right-Most Coil

At any instant, since all inductors are identical and in parallel, the total current divides equally among them.

$$ I_{coil} = \frac{I_{total}}{7} = \frac{I_1 + I_2}{7} $$