Sign Confusion Electric Potential

Question

I was solving a question and trying to find the potential difference between two points  and got confused with the signs . We know $$dV = -E.dr .$$ In this equation there is dot product between E and dr vector. Is it necessary to include magnitude of dr vector (Like dr can be positive or negative depending on whether r is increasing or decreasing) along with sign of dot product of E and dr vector or is it included along with the dot product.

Niescte · Answer

This is a very common doubt which arises in electrostatics and gravitation when we try to derive formulas by bringing a charge or mass(m) from infinity to to a certain point. Most books and teachers derive it the other way from a certain point to infinity. But whenever student try to derive it the other way they are left confused.
The reason is that whenever use this formula with integration, you have to check sign twice. Once while aplplying dot product and second while applying dr. dr can be positive or negative depending on whether r is increasing or decreasing. Most students ignore the second point and arrive at wrong results.
To understand the intuition and reasoning behind this see : https://physics.stackexchange.com/a/626556/293064

my2cts · Answer

$E$ and $mathrm dr$ are vectors.  Vectors do not have a sign. They have a direction and a magnitude. It is the dot product of these that has a sign.

Sign Confusion Electric Potential

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