Gradient, Electric Potential, and Electric Field

[KvGroOve]

Hi guys!

I was wondering about the relation between the Gradient, Electric Potential, and Electric Field. I know that if you take the Gradient of a scalar field, you get a resultant vector field in which the vector points in the direction of greatest increase when you take a infinitesimally small step.

1) What if there are multiple directions which all offer the greatest yet same amount of increase in the value of the scalar function? I believe that if it's a local minimum, the gradient of that specific spot is 0. But what if it's not a local minimum? How does the gradient operator handle that sort of situation? (I'm guessing that would mean a cross-section of the scalar field would trace out an inflection point).

2) The Electric Field can be given by

E=-∇φ​

So from my understanding, the negative sign just makes the vector point in the opposite direction, which in this case, is the direction opposite to the greatest increase of Electric Potential. Is this direction guaranteed to be the direction of the greatest possible decrease of the Electric Potential? If so, it's not intuitive to me that the direction of greatest possible decrease is opposite of the direction of greatest possible increase. And again, what if there are multiple directions that offer the greatest possible decrease in Electric Potential?

Thanks!

 

[Orodruin]

1) What if there are multiple directions which all offer the greatest yet same amount of increase in the value of the scalar function? I believe that if it's a local minimum, the gradient of that specific spot is 0. But what if it's not a local minimum? How does the gradient operator handle that sort of situation? (I'm guessing that would mean a cross-section of the scalar field would trace out an inflection point).

This does not happen. As long as the function can be well approximated by a linear function, there will always be a unique direction of fastest increase.

If the gradient is zero, then the linear part of the change in the function value vanishes and the change in the function is of second order or higher in the displacement.

2) The Electric Field can be given by
E=-∇φSo from my understanding, the negative sign just makes the vector point in the opposite direction, which in this case, is the direction opposite to the greatest increase of Electric Potential. Is this direction guaranteed to be the direction of the greatest possible decrease of the Electric Potential? If so, it's not intuitive to me that the direction of greatest possible decrease is opposite of the direction of greatest possible increase. And again, what if there are multiple directions that offer the greatest possible decrease in Electric Potential?

It is a linear approximation of the function near the point so yes, it is the direction of fastest decrease in potential. Again, this presupposes that the function can be approximated by a linear function near the point (think a plane in 3D). If this is not the case the function is not differentiable.