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From chapter $7$ of Kandel et all's Principles of Neural Science:, Go to page 12, Box 7-2 (particular image attached below)

The author claims that $$V_m = E_{Na} + I_{Na} / g_{Na}$$ on one side of the equation and then says $$V_m = E_{K} + I_{K} / g_{K}$$ by looking at the other side of the circuit.

But how can this be? Since the battery is showing a negative side facing $V_{in}$ doesn't it have to be

$$V_m = -E_{K} + I_{K} / g_{K}$$

Am I misunderstanding something or did the author make a mistake in calculation?

enter image description here

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The author has just chosen to draw the battery "backward" in the diagram to go along with the negative sign. In the equations, the direction of the battery should just be oriented by convention according to outward vs. inward.

Note that $E_{K} = -75 mV$. If you were to use instead your equation:

$$V_m = -E_{K} + I_{K} / g_{K}$$

you would need to take $E_{K} = 75 mV$ and you'd arrive at the same answer.

I'd say there is a bit of a physics vs. biology clash here; in physics, it's more typical to go with certain explicit/literal conventions which then fit nicely into particular equations, and that's how you're thinking about it. In biology, we tend to take some shortcuts.

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