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So, I'm having a bit of trouble understanding exactly WHY resistances add in series. My confusion comes when we compare circuits to water and resistances to the cross-sectional area of the tubes carrying the water. Tubes of varying widths

I would assume that the water that flows through the tubes would only be dependent on the tube of the smallest width, since the water that can flow through the smallest tube can also fit in the larger tubes. So, I am a bit confused as to why these resistances would add instead of the equivalent resistance just being equal to the maximum value resistor we have.

I know this is a basic question, but I truly do feel that understanding the why is just as important as understanding the what. I want to know both, and Physics StackExchange seems like a great place to ask this.

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  • $\begingroup$ Your intuition with water is wrong. Imagine how a wide but very long channel would affect flow. $\endgroup$
    – user137289
    Commented Feb 15, 2018 at 7:26
  • $\begingroup$ It sounds like you're assuming that there is air in the pipes, which is not allowed in this analogy. Also, FWIW, analogies are useful for building mental pictures, but they aren't to be taken too seriously. The real answer to your question is given by Whit3rd - it's a direct consequence of Ohm's law, and the fact that for resistors in series, the current is the same through all of them. $\endgroup$
    – J. Murray
    Commented Feb 15, 2018 at 8:05

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Resistance, by definition, is a voltage divided by a current. So, a series of resistors has a terminal voltage which is the sum of the voltage drops of the individual resistors, and (because the resistors are in series) an operating current which is identical for all those series elements.

The formula for the resistance of the ensemble is: $$ R_{123} = {V_1 + V_2 + V_3 \over {I_{123}}} = V_1 / I_{123} + V_2 / I_{123} + V_3 / I_{123} = R_1 + R_2 + R_3$$

The water analogy, alas, is to a NONLINEAR kind of flow reduction, and the 'smallest aperture dominates' model that might work for fluids is not accurate in dealing with the linear electrical resistance situation. Potential drop in a liquid has quadratic behavior at high currents, not linear in current.

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