The relationship between electric potential and field.

In summary, the electric potential at the centre of a circle with 12 evenly spread electrons is 12V, as electric potential is a scalar and can be found using the formula V=kq/R. However, the electric fields in both situations are different, as the field is related to the gradient of the potential and not the potential itself. This means that even if the potential has the same value at a point, the gradient can be different, resulting in different electric fields. Therefore, even though the two situations have the same potential, the electric fields are different due to the varying slopes of the potential.
  • #1
Yumrey
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Homework Statement


There are 12 electrons spread on the circumference of a circle with radius R evenly. What's the electric potential at the centre? Then the electrons are concentrated on the upper half of the circle, spread evenly. What's the electric potential at the centre now?

Homework Equations

The Attempt at a Solution


Since electric potential is a scalar, the potential due to one electron is V=kq/R. By superposition principle, the potential due to all the electrons is simply 12kq/R=12V. Both situations have the same potentials.
What confuses me is that the electric fields in both situations are indeed different. In the first case, the field is simply zero because they all cancel at that point, while it's nonzero in the second case. But electric field is the negative gradient of the electric potential, then how come the same potentials result in different fields?
 
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  • #2
The field is related to the gradient. The same numerical value at a point may not have the same gradient.

Gradient is like a two or three dimensional slope.

Two functions can have the same value at a point but have different slopes.
 
  • #3
Because the field is proportional to the gradient of the potential and not the potential. Also in one dimension you can have functions which take the same value in a point but have different derivatives.
 
  • #4
To expand on that, the field depends on how the potential varies in a neighbourhood of a point, not on the value in the point only.
 
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  • #5
Dr. Courtney said:
The field is related to the gradient. The same numerical value at a point may not have the same gradient.

Gradient is like a two or three dimensional slope.

Two functions can have the same value at a point but have different slopes.

But then how should I differentiate the two Vs(which have the exact same expression) such that they'll yield different electric fields?
 
  • #6
Think of potential like the height of the ground at a point. If the ground is flat, a ball won't roll. If the potential is flat, a charge won't move because there is no net electric force.

But if the ground is sloped, the ball will roll. If the potential is sloped, a charge will accelerate, because there is a net electric force.
 
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  • #7
Yumrey said:
But then how should I differentiate the two Vs(which have the exact same expression) such that they'll yield different electric fields?
Read my last post before this one. If you only know the function value in a single point you cannot take the derivative.
 
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1. How are electric potential and electric field related?

The electric potential at a point in space is directly proportional to the electric field at that point. This means that the strength of the electric field determines the amount of electric potential at a specific location.

2. What is the difference between electric potential and electric field?

Electric potential is a scalar quantity that measures the amount of electric potential energy per unit charge at a point in space. Electric field, on the other hand, is a vector quantity that measures the strength and direction of the force on a charged particle at a point in space.

3. How is the electric field calculated from the electric potential?

The electric field can be calculated by taking the negative gradient of the electric potential. This involves finding the change in electric potential in all three dimensions and then taking the negative of each of those changes.

4. What is the unit of measurement for electric potential and electric field?

Electric potential is measured in volts (V), while electric field is measured in volts per meter (V/m).

5. How does the electric potential change with distance from a point charge?

The electric potential decreases as the distance from a point charge increases. This is because the electric field also decreases with distance, and the two are directly related. The farther away you are from a point charge, the weaker the electric potential will be.

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