Reading:

Read B & O Section 1.9; KB 361 lecture notes 9 -- 11.

Problems for Review:

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1. B & O 1-24.
2. B & O 1-26.
3. Consider an LC circuit, like that pictured in Figure 1-8 with R=0.
   1. Write the differential equation for q(t), the charge on the capacitor. What is this ODE's general solution?
   2. What is q(t) on the capacitor, if it initially carries charge , with no current flowing?
   3. Now consider the LC circuit with a voltage source, of constant voltage (that is, Figure 1-9 with R=0 and ). Write the differential equation for q(t), and find its particular solution.
   4. For the LC circuit in part (c), if the capacitor is initially discharged, with no current flowing (q(0) = i(0) = 0), find q(t) on the capacitor.
   5. The particular solution is often called the steady state solution, and the homogeneous solution the transient solution. Comment on whether that nomenclature makes sense in the case of an LC circuit.

4. Consider an RC circuit whose voltage source carries voltage (that is, Figure 1-9 with L=0 and ). Write the differential equation for q(t), the charge on the capacitor. Using the approach of B & O section 1-9, find a particular solution of the form

   

   (That is, find and .) Show that the amplitude of the response is maximal when . Plot the square of the magnification factor, , and the phase lag .

Problems to Solve:

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1. B & O 1-20.
2. B & O 1-21. HINT: All the work has been done for you in B & O section 1-9, you just need to figure out how to extract the result you need.
3. B & O 1-23.
4. Consider an RC circuit, like that pictured in Figure 1-8 with L=0.
   1. Write the differential equation for q(t), the charge on the capacitor. What is this ODE's general solution?
   2. What is q(t) on the capacitor, if it initially carries charge ?
   3. Now consider the RC circuit with a voltage source, of constant voltage (that is, Figure 1-9 with L=0 and ). Write the differential equation for q(t), and find its particular solution.
   4. For the RC circuit in part (c), if the capacitor is initially discharged (q(0) = 0), find the charge q(t) and current on the capacitor.
   5. Note the values of q(t) and initially (t=0) and asymptotically ( ). Explain with reference to Kirchoff's law, how the circuit evolves, yet always distributes the voltage drop over the capacitor and resistor.
   6. The particular solution is often called the steady state solution, and the homogeneous solution the transient solution. Comment on whether that nomenclature makes sense in the case of an RC circuit.

5. Consider an LC circuit whose voltage source carries voltage (that is, Figure 1-9 with R=0 and ).
   1. Write the differential equation for q(t), the charge on the capacitor. Using the approach of B & O section 1-9, find a particular solution of the form

      

      (That is, find .)

   2. For this LC circuit, if the capacitor is initially discharged, with no current flowing (q(0) = i(0) = 0), find q(t) on the capacitor.
   3. Show that, in the limit where , where , the charge q(t) on the capacitor approaches

      

      HINT: You could do this with trig identities, or by Taylor expanding about the argument .)

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