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