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Question Video: Determining Capacitance for Capacitors in Parallel Physics

Two capacitors, 𝐢₁ and 𝐢₂, are connected in parallel, where 𝐢₁ > 𝐢₂. Which of the following statements correctly relates the total capacitance, 𝐢_total, to 𝐢₁ and 𝐢₂? [A] 𝐢₂ < 𝐢_(total) < 𝐢₁ [B] 𝐢_(total) = (𝐢₁ + 𝐢₂)Β² [C] 𝐢_(total) = 𝐢₁𝐢₂ [D] 𝐢₁ = 𝐢_(total) βˆ’ 𝐢₂ [E] 𝐢_(total) = (𝐢₁/𝐢₂) + (𝐢₂/𝐢₁)

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Video Transcript

Two capacitors, 𝐢 one and 𝐢 two, are connected in parallel, where 𝐢 one is greater than 𝐢 two. Which of the following statements correctly relates the total capacitance, 𝐢 total, to 𝐢 one and 𝐢 two? (A) 𝐢 two is less than 𝐢 total, which is less than 𝐢 one. (B) 𝐢 total equals the square of 𝐢 one plus 𝐢 two. (C) 𝐢 total equals 𝐢 one times 𝐢 two. (D) 𝐢 one equals 𝐢 total minus 𝐢 two. (E) 𝐢 total equals 𝐢 one over 𝐢 two plus 𝐢 two over 𝐢 one.

This question is asking us about two capacitors connected in parallel. So, we could imagine a circuit like this, where we have the two capacitors, 𝐢 one and 𝐢 two, on separate parallel branches. We’re being asked how the total capacitance, 𝐢 total, relates to these individual capacitances, 𝐢 one and 𝐢 two. To work this out, it will be helpful to remember that the total capacitance for a parallel combination of capacitors is given by 𝐢 total equals 𝐢 one plus 𝐢 two plus 𝐢 three and so on. That is, for capacitors connected in parallel, we add together the individual capacitances to find the total capacitance of the combination.

In this case, we have just two capacitors with capacitances 𝐢 one and 𝐢 two. So, this relationship would become 𝐢 total equals 𝐢 one plus 𝐢 two. Now, since 𝐢 total is equal to the sum of 𝐢 one and 𝐢 two, we can discount answer option (A), which claims that 𝐢 total is larger than 𝐢 two but smaller than 𝐢 one. If 𝐢 total is the sum of the two individual capacitances, then it must be larger than both 𝐢 one and 𝐢 two. Obviously, then, option (A) cannot be correct.

To determine which of the four remaining options is correct, let’s rearrange this equation by subtracting 𝐢 two from each side. This gives us an equation that says 𝐢 one equals 𝐢 total minus 𝐢 two. Looking at the remaining answer options we are given, we can see that our equation matches the one given in option (D). We have found then that the correct answer is given in option (D). The statement that correctly relates the total capacitance 𝐢 total to the individual capacitances 𝐢 one and 𝐢 two is 𝐢 one is equal to 𝐢 total minus 𝐢 two.

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