- #1

zenterix

- 689

- 83

- Homework Statement
- I am stuck on the following problem.

- Relevant Equations
- Determine an expression for ##V_C##, the voltage across the capacitor.

Actually, I would like to solve the problem for every variable.

Here is the circuit as it appears in MIT OCW Vibrations and Waves Problem Solving course

Here is my own picture

I wrote equations for the three loops.

$$-V_0\sin{\omega t}+IR+\frac{Q}{C}=0\tag{1}$$

$$-V_0\sin{\omega t}+IR=-L\dot{I}_2\tag{2}$$

Only two of these equations are independent. We can obtain the third equation given any two of the equations.

We also have

$$I=I_1+I_2\tag{4}$$

$$I_1=\dot{Q}\tag{5}$$

I am really not sure what to do at this point. I've tried to manipulate the equations above to get a differential equation just in one of the variables but I cannot. For example

$$-V_0\sin{\omega t}+(\dot{Q}+I_2)R+L(\dot{I}-\ddot{Q})=0\tag{6}$$

$$\ddot{Q}-\frac{R}{L}\dot{Q}-\frac{R}{L}I_2=-\frac{V_0}{L}\sin{\omega t}+\dot{I}\tag{7}$$

I would very much appreciate some help here.

Here is my own picture

I wrote equations for the three loops.

**Outer Loop Through ##C##**$$-V_0\sin{\omega t}+IR+\frac{Q}{C}=0\tag{1}$$

**Outer Look Through ##L##**$$-V_0\sin{\omega t}+IR=-L\dot{I}_2\tag{2}$$

**Small Inner Loop**

$$\frac{Q}{C}=L\dot{I}_2\tag{3}$$$$\frac{Q}{C}=L\dot{I}_2\tag{3}$$

Only two of these equations are independent. We can obtain the third equation given any two of the equations.

We also have

$$I=I_1+I_2\tag{4}$$

$$I_1=\dot{Q}\tag{5}$$

I am really not sure what to do at this point. I've tried to manipulate the equations above to get a differential equation just in one of the variables but I cannot. For example

$$-V_0\sin{\omega t}+(\dot{Q}+I_2)R+L(\dot{I}-\ddot{Q})=0\tag{6}$$

$$\ddot{Q}-\frac{R}{L}\dot{Q}-\frac{R}{L}I_2=-\frac{V_0}{L}\sin{\omega t}+\dot{I}\tag{7}$$

I would very much appreciate some help here.