### Video Transcript

Which of the following graphs
represents the ratio of the current going through resistors A, B, and C in the given
figure? Graph (A), graph (B), graph (C), or
graph (D).

To work out the ratio of the
currents going through each resistor, we need to recall how the current splits
across each parallel path. The total current in a circuit with
parallel components is given by the rule ๐ผ total equals ๐ผ one plus ๐ผ two plus ๐ผ
three and so on. That is, the total current splits
along all the parallel branches. In this question, we have three
different branches A, B, and C, so this rule becomes ๐ผ total equals ๐ผ A plus ๐ผ B
plus ๐ผ C.

We can also recall that the
potential difference across each branch of a parallel circuit is the same, so ๐ A
equals ๐ B equals ๐ C. That is, the potential difference
across each of these three resistors is the same. Since the three potential
differences are the same, we will say that the potential difference across each
branch is equal to ๐. So we know about the current and
the potential difference for each parallel branch. Weโre also given the resistances
for each resistor. Resistor A has resistance two ๐
,
resistor B has resistance three ๐
, and resistor C has resistance ๐
.

We can now use Ohmโs law along each
path to work out the currents in each path and then compare the currents to see
which of the graphs correctly represents the ratio of the current going through the
resistors. Ohmโs law can be written as ๐
equals ๐ผ times ๐
, where ๐ is the potential difference, ๐ผ is the current, and ๐
is the resistance.

Weโre concerned with finding the
current, so letโs rearrange the equation to make the current, ๐ผ, the subject. We can do this by dividing both
sides of the equation by the resistance ๐
. The resistances on the right-hand
side cancel each other, and we are left with an equation that says the current ๐ผ
equals the potential difference ๐ divided by the resistance ๐
. Now we can use this equation to
work out the current along each path by substituting in the relevant values.

For path A, we have ๐ผ A equals ๐
divided by two ๐
. For path B, we have ๐ผ B equals ๐
divided by three ๐
. For path C, we have ๐ผ C equals ๐
divided by ๐
. We can rewrite these first two
equations, factoring out the numerical values. Then, from these three equations
for the currents ๐ผ A, ๐ผ B, and ๐ผ C, we can see that the current through resistor
C is the largest, the current through resistor B is the smallest, and the current
through resistor A is between these two values.

We can express that as the
inequality ๐ผ C is greater than ๐ผ A, which in turn is greater than ๐ผ B. Looking at each of the graphs, we
can see that the graph in option (A) matches the ratio of currents that weโve just
calculated. Therefore, the correct answer is
option (A). The graph in option (A) correctly
represents the ratio of the current going through the resistors A, B, and C in the
given figure.