electrical-resistanceelectricityhomework-and-exercises
Given a setup like:
Is it possible to find current and voltage through each resistors given resistor values and overall voltage (battery voltage I suppose).
I think $$I_0=I_A=\frac{\Delta V}{R_{total}}$$
I know $I_B+I_C=I_A$ but how do I find $I_B$ if I don't have $\Delta V_B$?
The diagram shows first four resistors in series then four resistors in parallel.
For the resistors in series the current flowing into the wire, $I_{in}$ must be the same as the current flowing out, $I_{out}$ because the current can't escape from the wire. There is only one route for the current to flow through the wire so the current has to pass through all the resistors in turn. That's why the current passing through every resistor must be the same.
Now look at the resistors in parallel. The point here is that the top ends of the resistors are all connected together so they must all be at the same voltage $V_{in}$. Likewise the bottom ends are all connected together so they must be at the same voltage $V_{out}$. That means all the resistors have the same voltage drop across them of $V_{in} - V_{out}$.
The voltage across either horizontal resistor is zero so they can be removed from the circuit without changing the solution.
This is most easily seen by simply removing the two horizontal resistors and then it's clear that the nodes the horizontal resistors connect to each have the same voltage. Thus, by Ohm's law, there is no current through either horizontal resistor since there is no voltage across either resistor.
In other words, it doesn't matter if the horizontal resistors are there or not so they can be removed without changing the solution.
You are then left with 3 identical paths with resistance 2R each.
Of course, in a real circuit, the real resistors will never be identical so, this ideal solution is only an approximation.
Best Answer
Two formulas to understand and internalize:
1) Voltage divider: For two resistors R1 and R2 in series, the voltage $V_2$ across $R_2$, relative to the total voltage $V_T$ across both resistors, is: $$\frac{V_2}{V_T}=\frac{1}{1+R_1/R_2} $$ This formula is a direct consequence of the fact that the current through the two series resistors is the same.
2) Current divider: For two resistors R1 and R2 in parallel, the current $I_2$ through $R_2$, relative to the total current $I_T$ through both resistors, is: $$\frac{I_2}{I_T}=\frac{1}{1+R_2/R_1} $$ This formula is a direct consequence of the fact that the voltage across the two parallel resistors is the same.
You can always check that you have the correct ratio by letting one resistor get very large relative to the other; for example, in the current divider, as $R_2/R_1$ goes to $\infty$, you expect the current through $R_2$ to go to 0.