Resistors and capacitors in circuits: series, parallel, etc.

When you have two resistors, one after another, in a circuit with a battery, the current through each resistor is the same, and that current is given by V/(R1+R2). (That is called series because the current goes through one element at a time, one after the other.)    To understand that, it might be useful to sketch a picture and to imagine the current, like a model train, going around the circuit first through one resistor, then through the other…

When you have two resistors on two different branches of a circuit,  current coming into the dividing point can either go into one branch or the other. Like a fork in the road, each car ( electron)  will either go on the left fork or the right fork.   If the two resistors are equal, the current will divide equally with half going through the left side of the circuit, and half taking the right fork and going through the right-hand side of the circuit. If resistor 1 has a higher resistance than resister 2, then more current will go through resistor 2.   The ratio, the way in which the current divides, is R1/R2.  So if R2  is twice as big as R1,  than the ratio is 2 to 1; which means that 1/3  of the current will go through R2 and 2/3 through R1.

To evaluate more complex circuits with resistors, one generally uses formulas which calculate effective resistances for resistors combined in complex ways. I won't ask you about that (on any test). No one remembers those formulas six months after the class anyway. We will focus on the concepts, and on simple circuits. Focusing on simple circuits takes us away from rote learning and application of formulas.

Simple circuits, however, are not necessarily easy, especially when there are capacitors involved. Capacitors tend to be more interesting than resistors, partly because they change with time. In a nutshell, resistors "dissipate" energy, and capacitors store energy.  To understand the nature of circuits which include capacitors, one ideally should have an understanding of current as a function of time in a circuit, and how the flow of current leads to the build-up of charge.

In all the problems we will consider that involve capacitors, there will be a gate (real or imaginary) and when you close the gate (turn on the switch), current will either flow in or out of the capacitor. (If it is charged to start with, current will flow out. If it is not charged to start with, current will flow in.)  Then, after a while, when the capacitor,  or really the circuit, reaches equilibrium.  In equilibrium no current ( zero)  will flow through the capacitor.  That is the key  to understanding capacitors in equilibrium. It is like they weren't there in the sense that no current flows through them.

In equilibrium,  or actually, steady state,  zero current flows through a capacitor. 

The time-dependent approach to that equilibrium state is generally exponential, just like we discussed in class. Sometimes in circuits that are a little complicated it is hard to know exactly what the exponential time dependence is, but it is always something like exp{-t/RC}, where R has units of resistance and C has units of capacitance.

Also, for an uncharged capacitor, in the beginning-- when the switch is first closed-- current will flow freely in and out of the capacitor. So, for a moment, the capacitor acts like a short-circuit––like a segment with no (zero) resistance. That is just for an instant. Then the current starts to decrease exponentially.  This concept, which seems less obvious to me, helps us understand the t=0 point on the graph of current versus time.