Is Electric Field and Potential opposite in directions to each other? Does the -ve sign in the formula
$$\ V_{ba}=V_b-V_a=-\int_a^b\vec{E}\cdot d\vec{l}$$
imply the Potential and Field are in opposite directions?
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$\begingroup$ Electric field is a vector field quantity so at each point it has magnitude (or length) and direction. While the electric potential is a scalar quantity. $\endgroup$– SG8Commented May 28, 2021 at 14:32
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$\begingroup$ there is a dot product between the electric field and the infinitesimal length, so you get a scalar value for the potential. $\endgroup$– N A McMahonCommented May 28, 2021 at 14:58
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3 Answers
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Potential does not have a direction. It's like a hight. You go "uphill" when going in the opposite direction to the direction of the ${\bf E}$ field. Just remember that its hard to climb a hill and easy to slide down.
answered May 28, 2021 at 14:29
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$\begingroup$ Also, adding to the above picture: If you are standing at a point on hilly, uneven terrain and look around you, the direction with the most decrease in height will be the direction of the "field". Now this by definition , definitely has a direction. $\endgroup$– SidarthCommented May 28, 2021 at 15:28
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Only the electric field can have a direction because it's a vector. Potential isn't a vector, in fact, as you wrote, in the integral you have a "dot product" between two vectors: $\vec{E}$ and $d\vec{l}$. The dot-product in real field is a bilinear shape positive defined that takes vectors and gives a real number.
The sign "-" is given only by definition of Potential.
In fact, electric potential from $a$ to $b$ is defined as: $$V_a-V_b=\int_a^b\vec{E} \cdot d\vec{l}$$
Or equivalently $$\Delta V=V_b-V_a=-\int_a^b\vec{E}\cdot d\vec{l}$$
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The potential difference $V$ between two points is the work per unit charge required to move the charge between the points. Work is a scalar quantity having magnitude only.
The direction of the electric field is, by convention, the direction of the force that a positive charge would experience if placed in the field. The electric field is a vector quantity have direction and magnitude.
So it does not make sense to talk about the "direction" of the field and the potential being opposite. On the other hand, the direction of the electric field corresponds to a decrease in the electrical potential of a positive charge moving in that direction. This is the basis of the minus sign of your equation.
Hope this helps.
