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Question Video: Determining the Ratio of Current Going through Different Resistors in a Parallel Circuit Physics • 9th Grade

Which of the following graphs represents the ratio of the current going through resistors A, B, and C in the given figure? [A] Graph A [B] Graph B [C] Graph C [D] Graph D

03:06

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 ๐ผ total equals ๐ผ one plus ๐ผ two plus ๐ผ 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 ๐ผ total equals ๐ผ A plus ๐ผ B plus ๐ผ 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. Weโ€™re 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 Ohmโ€™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. Ohmโ€™s law can be written as ๐‘‰ equals ๐ผ times ๐‘…, where ๐‘‰ is the potential difference, ๐ผ is the current, and ๐‘… is the resistance.

Weโ€™re concerned with finding the current, so letโ€™s rearrange the equation to make the current, ๐ผ, 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 ๐ผ 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 ๐ผ A equals ๐‘‰ divided by two ๐‘…. For path B, we have ๐ผ B equals ๐‘‰ divided by three ๐‘…. For path C, we have ๐ผ 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 ๐ผ A, ๐ผ B, and ๐ผ 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 ๐ผ C is greater than ๐ผ A, which in turn is greater than ๐ผ B. Looking at each of the graphs, we can see that the graph in option (A) matches the ratio of currents that weโ€™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.

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