in this video we're given the total current flowing through a circuit and then the main goal
of the problem is to find this unknown resistance capital r in the left branch of the circuit we're
given an ideal emf of four and a half volts that's in series with an internal resistance
of 1 ohm and that makes the potential difference between nodes a and b the terminal voltage of
the power source so in part a that's what we're finding in part b we'll find the current through
the central branch of the circuit and then in part c we'll finally compute this unknown resistance so
to get the terminal voltage of the battery if i'm being formal about this i can say that va
the potential at node a plus the four and a half volts for the ideal emf minus the voltage drop
across the internal resistance well that's given by ohm's law it's i times r that's all equal to
the potential at node b then what we mean by this notation v a b is actually the difference v b
minus v a so i subtract v a from both sides and then simplify this expression 4.5 minus 0.784
times 1. when i run the numbers on this i get a terminal voltage of 3.716 volts now it's not
necessary to be that formal about computing the terminal voltage i could just say well here's my
ideal power source four and a half volts minus the drop over the internal resistance which is
given by ir so just four and a half minus ir in part b we want to get the current through the 10
ohm resistor and we have the potential difference across that resistor because the voltage at the
high side of that 10 ohm resistor is the same vb and at the low side i'm connected to ground
but just more generally speaking that's my va because it's connected directly by wire to node
a so i have the same exact potential difference of 3.716 volts across that 10 ohm resistor
i apply ohms a lot of this i'll write it as i equals v over r and i have a 3.716 volt potential
difference over a 10 ohm resistor which gives me a current of 0.3716 amps now the whole reason this
is useful is that it allows me to compute how much current is flowing in the right branch of
the circuit we had a total of 0.784 amps flowing in this circuit we just figured out how much flows
down this middle branch when the current splits that's 0.3716 amps and the leftover amount is the
amount flowing through the right branch i'll just call that ir so ir is just the total amount of
current minus the amount that went down the center branch and this gives me 0.4124 amps flowing down
that right branch now the reason this is useful for finding our unknown resistance r is that i
have the potential difference across that right branch it's that same 3.716 volts because again
we're just directly connected by wire to nodes a and b at the top and bottom of this branch so
i have the potential difference across it i have the total current through it that allows me to get
the total resistance of that branch and then if i just subtract the 5 ohms out of that i'll have
the resistance capital r so working through it somewhat formally i'm going to use ohm's law for
this v equals ir and the potential difference across this symbolically is v-a-b and the total
current through it i call that i sub r so i'll write it symbolically for a second and then my
total resistance is going to be capital r plus the 5 ohm resistor so dividing by the current ir
and then subtracting the 5 ohms i can solve for my capital r and we plug the numbers in
for the terminal voltage that was 3.716 and the current in the right-hand branch that was 0.4124
and we're going to subtract 5 ohms from that at the end and i get a resistance of 4.01 ohms
and that's it if you enjoyed this video or at least found it useful
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