Question

Initially the capacitor is uncharged, but after the switch is closed, it quickly starts charging.

(a) UsingKirchhoff’sLaws,writedownadifferentialequationfor?(?).

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(b) Using the method “separation of variables” to solve the differential equation of part (a), and show that the solution can be written as ?(?) = ?????1 − ???/??

(c) Determine an expression for time constant ? in terms of ?, ??, ??, and ?.

4. (d) Determine an expression for ???? in terms of ?, ??, ??, and ?.

5. (e) If resistor ?? wasn’t there, we would have the usual RC circuit with just one R and one C. Take the limit ?? → ∞ and determine the time constant ? for this special case. Does the result match what you expected?

PROBLEM 2

Consider the circuit on the right.

The goal in this problem is to determine the direction and magnitude of the current through each battery.

(a) UseKirchhoff’sLawstowritedownasystemofthreeindependentequationsthatrelate the three currents in this circuit.

2. (b) Convertyoursystemofequationsintoamatrix.

3. (c) Use Gaussian elimination to convert your matrix to “row echelon form”.

(d) Determine the current through each battery. Don’t forget your units!

??? = _____________________ ____ ??? = _____________________ ____ ??? = _____________________ ____

[ ] going right [ ] going right [ ] going right

[ ] going left [ ] going left [ ] going left

[ ] is zero [ ] is zero [ ] is zero

Answer 1