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 𝐼 total equals 𝐼 one plus 𝐼 two plus 𝐼
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 𝐼 total equals 𝐼 A plus 𝐼 B
plus 𝐼 C.
We can also recall that the
potential difference across each branch of a parallel circuit is the same, so 𝑉 A
equals 𝑉 B equals 𝑉 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 𝑉. So we know about the current and
the potential difference for each parallel branch. We’re also given the resistances
for each resistor. Resistor A has resistance two 𝑅,
resistor B has resistance three 𝑅, and resistor C has resistance 𝑅.
We can now use Ohm’s 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. Ohm’s law can be written as 𝑉
equals 𝐼 times 𝑅, where 𝑉 is the potential difference, 𝐼 is the current, and 𝑅
is the resistance.
We’re concerned with finding the
current, so let’s rearrange the equation to make the current, 𝐼, the subject. We can do this by dividing both
sides of the equation by the resistance 𝑅. The resistances on the right-hand
side cancel each other, and we are left with an equation that says the current 𝐼
equals the potential difference 𝑉 divided by the resistance 𝑅. 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 𝐼 A equals 𝑉
divided by two 𝑅. For path B, we have 𝐼 B equals 𝑉
divided by three 𝑅. For path C, we have 𝐼 C equals 𝑉
divided by 𝑅. We can rewrite these first two
equations, factoring out the numerical values. Then, from these three equations
for the currents 𝐼 A, 𝐼 B, and 𝐼 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 𝐼 C is greater than 𝐼 A, which in turn is greater than 𝐼 B. Looking at each of the graphs, we
can see that the graph in option (A) matches the ratio of currents that we’ve 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.
