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2

The minus sign is only there by convention. You could replace $\phi$ with $-\phi$ and the minus sign would go away. Note that $\nabla \phi$ points in the direction of steepest ascent for $\phi$, whereas $-\nabla \phi$ points in the direction of steepest descent. Perhaps it seems nice for the force on an object to be pointing in a direction of descent for ...

2

It's total convention. The idea is that one can caluclate, for any path: $$\Delta\left(\frac{1}{2}mv^{2}\right) = \int_{\rm path}{\vec F}\cdot d{\vec s}$$ One can then split the left hand side into a conservative bit and a nonconservative bit. (with the definition of conservative being ${\vec \nabla}\times{\vec F} = 0$). Then, since we know that any ...

-2

We take positive charge as a test charge because positive charge is higher potential and negative charge is lower potential. Therefore, influence of positive charge on other charges is greater than negative charges. We can also take negative charge but the effect will be lower.

0

As that wikipedia link suggests, gauge pressure is compared to the surrounding air pressure. It really is intended to be used for "what a pressure gauge reads" in "everyday life". The all-surrounding presence of air on the Earth makes it a convenient zero point, so it is immediately obvious when things are not in equilibrium with the environment. Your ...

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