in this video we're presented with a resistor network that has resistors that are both in
parallel and in series and we're told that the thing we actually measure in the lab
is the potential difference between two points in the circuit so we're going to simplify and solve
this resistor network and that means we're going to find the voltage across and the current through
every single resistor and then we're going to use those results to find the potential differences
between a bunch of pairs of nodes so we begin the simplification by noticing that we have
two resistors that are clearly in parallel here they're directly connected by wire at their high
end and at their low ends so they're guaranteed to always have the same voltage across them
which is the key to parallel resistors so there's our parallel equivalent resistance formula
i plug the numbers into my calculator 1 over 220 plus 1 over 100 press enter and then 1 over that
result gives me the reciprocal i'm looking for and i get 68.8 ohms now we redraw the circuit with the
equivalent resistor put in there now i'm left with a circuit that has four resistors that are clearly
in series because the same current must pass through every one of these there's no branching in
the circuit anymore so i write down my equivalent resistance for series resistors formula all you do
is add them and when i add them up i get 315.8 ohms and we redraw the circuit one last time
and now that i'm down to one equivalent resistor i can find the total current flowing in the circuit
i equals v over r 3 volts 315.8 ohms and i get a total current of 0.00950 amps which
i'm going to write as 9.50 milliamps now i start to work backwards through the circuit
and the simplification i did at this last step was a group of series resistors and
what they share in common is the same current so every one of them has 9.50 milliamps of current
now that i have the current through each of those resistors i can find the voltage drop over
each of them using ohm's law v equals ir i'm just showing one sample calculation in here for the 47
ohm resistor i take .0095 amps and multiply by 47 ohms when i do that i get a voltage drop of 0.447
volts i do the same thing for the rest of the resistors i get 0.950 volts for the first 100
ohm resistor and since the other 100 ohm resistor has the same current flowing through it that must
also be 0.950 volts and then across the equivalent resistance that came from the parallel combination
i have 0.654 volts again just multiplying the current and the resistance to get that then i work
back to my original circuit and what i did in the first step was combine the two parallel resistors
into a single equivalent resistor and what those two resistors in parallel share is the same
voltage drop across them so the voltage dropped from here to here is 0.654 volts and that's going
to be true for each of the individual resistors as well so each of these has 0.654 volts across it
that allows me to solve for the current in each of those resistors using ohm's law again
i'm going to write i equals v over r and for the left resistor in the parallel combination
i get .00297 or 2.97 milliamps for the resistor on the right in the parallel combination i get 6.54
milliamps and i can do a quick check on my work here when i add those two currents
i get to a total of 9.51 which is only slightly different from the 9.50 i calculated and that's
just due to rounding errors propagating through the calculations so that's good now i've actually
already got my complete solution because i have the voltage and current for every single resistor
but it might be nice to draw them all in the original circuit so i'll get that done real quick
so there's our completely solved resistor network and now i want to find the potential differences
between a bunch of nodes vab is just the voltage across the 47 ohm resistors so there's no extra
work to do here that's 0.447 volts likewise vbc is the voltage across the first 100 ohm resistor and
that's 0.950 volts and vcd you could take either path between c and d and either way you go you're
going to get .654 volts and then vac that requires that you jump over the 47 ohm resistor and
the first 100 ohm resistor those are both voltage drops because that's the direction
current is flowing and so you just add these two voltage drops and when i do that i get 1.40 volts
similarly the drop from b to d is given by the drop over this 100 ohm resistor added to the
drop over either one of the resistors that's in the parallel combination those are both the
same drop of 0.654 and when i do that i get 1.60 volts so this is how we make and test predictions
in the lab it's always potential differences because they're the easiest thing to measure
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