Electric field in a region with constant potential

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Homework Statement

When the potential is constant throughout a given region of space, is the electric field in the region also zero?

Homework Equations

in a parallel plate capacitor,
electric filed = - delta V / delta d

The Attempt at a Solution

It is difficult to understand how in a given region where the electric potential could be constant. That means at any given location, V=kq/r is the same everywhere and delta v equals zero. I can only relate this to a plate capacitor where the electric field is defined as - delta v / delta d. If delta v is zero, then E=0. Am I on the right track?

 

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Homework Statement

When the potential is constant throughout a given region of space, is the electric field in the region also zero?

Homework Equations

in a parallel plate capacitor,
electric filed = - delta V / delta d

The Attempt at a Solution

It is difficult to understand how in a given region where the electric potential could be constant. That means at any given location, V=kq/r is the same everywhere and delta v equals zero. I can only relate this to a plate capacitor where the electric field is defined as - delta v / delta d. If delta v is zero, then E=0. Am I on the right track?

yes. that equation you used comes from the calculus definition that relates electric potential and electric field.

$$E = -\frac{dV}{dr}$$

this equation simplifies to the one you wrote if the rate of change of voltage with respect to displacement is a constant.

So electric field equals the negative of the rate of change in voltage. If voltage is constant, meaning it has a rate of change of zero, electric field also is zero.