Does equal electric field imply equal potential?

In summary, the electric potential between points A and B on the z axis, where A is at +2m and B is at -4m, cannot be determined solely based on the fact that the electric field is the same at both points. This is because the electric potential depends not only on the electric field, but also on the distance between the points and the origin. However, if the electric potential is symmetric around the origin, as in this case, then points that are equidistant from the origin will have the same electric potential. The potential difference between A and B can be calculated using the formula V_B - V_A = -∫E·dl, and it is the same regardless of the direction of integration.
  • #1
fishingspree2
139
0

Homework Statement


Given the E field
E = 18/R2 R, R is the radial direction.
Find the electric potential between A and B where A is at +2m and B at -4m, both on the z axis.


The Attempt at a Solution


My question is, since E field depends only on R, the distance between the point and the origin, then E field is the same for point B at -4m and point C at +4m. Therefore they are equipotential. Therefore I can compute the potential difference between A at +2m and C at +4m. Is this reasoning correct?
 
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  • #2
No, equal electric field does not imply equal electric potential. The electric field is the gradient of the electric potential: E = -∇V, or in one dimension, E = -dV/dr.

So, if the electric field strength and direction are the same at two points, it doesn't mean that V is the same at those two points, it just means that the derivative of V is the same at those two points.

Example: electric field between two large parallel plates of uniform surface charge, is the same everywhere between the two plates, but electric potential increases linearly as you move from one plate to the other.

(The approximately uniform gravitational field close to the surface of the Earth is a totally analogous situation. The gravitational field strength is the same at two points at two different heights, but the gravitational potential is definitely different between those two points.)
 
  • #3
Hmm...what I said above was the most general answer. HOWEVER in this case, it seems like the problem posesses some symmetry. If dV/dr depends only on r, then V will depend only on r as well. So V is the same at all points that are equidistant from the origin.
 
  • #4
cepheid said:
Hmm...what I said above was the most general answer. HOWEVER in this case, it seems like the problem posesses some symmetry. If dV/dr depends only on r, then V will depend only on r as well. So V is the same at all points that are equidistant from the origin.
Some questions...
I'm trying to compute V between two general points for the given E field, just to see what it will give and also for the sake of it.
We know that

Vb-Va = [itex]-\int_{A}^{B}\overrightarrow{E}\cdot d\overrightarrow{l}[/itex]

any idea how I could express [itex]d\overrightarrow{l}[/itex]...?
also, should [itex]d\overrightarrow{l}[/itex] be from A to B or from B to A?
is Vb-Va called the potential difference between A and B or the potential difference between B and A?
 
  • #5
fishingspree2 said:
Some questions...
I'm trying to compute V between two general points for the given E field, just to see what it will give and also for the sake of it.
We know that

Vb-Va = [itex]-\int_{A}^{B}\overrightarrow{E}\cdot d\overrightarrow{l}[/itex]

any idea how I could express [itex]d\overrightarrow{l}[/itex]...?
also, should [itex]d\overrightarrow{l}[/itex] be from A to B or from B to A?
is Vb-Va called the potential difference between A and B or the potential difference between B and A?
I found that [itex]d\overrightarrow{l}=dR\overrightarrow{R}+Rd\phi \overrightarrow{\phi}+dz\overrightarrow{z}[/itex]

so
[itex]V_{B}-V_{A}-\int_{A}^{B}\overrightarrow{E}\cdot d\overrightarrow{l}=-\int_{R_{A}}^{R_{B}}\frac{18}{R^{2}}dR=\frac{18}{R_{B}}-\frac{18}{R_{A}}[/itex]
is that correct? in that case it seems my initial assumption that -4 and 4 are equipotential is correct since it depends only on R which is 4 in both cases.
is that the potential change when you go from B to A or is it the potential change when you go from A to B?
 
  • #6
fishingspree2 said:
I found that [itex]d\overrightarrow{l}=dR\overrightarrow{R}+Rd\phi \overrightarrow{\phi}+dz\overrightarrow{z}[/itex]

so
[itex]V_{B}-V_{A}-\int_{A}^{B}\overrightarrow{E}\cdot d\overrightarrow{l}=-\int_{R_{A}}^{R_{B}}\frac{18}{R^{2}}dR=\frac{18}{R_{B}}-\frac{18}{R_{A}}[/itex]
is that correct? in that case it seems my initial assumption that -4 and 4 are equipotential is correct since it depends only on R which is 4 in both cases.
is that the potential change when you go from B to A or is it the potential change when you go from A to B?

Obviously V_B - V_A is the potential change when you go from A to B. You're taking the difference between the final value and the initial value to compute the change.
 

1. What is an electric field?

An electric field is a physical quantity that describes the strength and direction of the force that an electric charge would experience at a specific point in space.

2. How is an electric field created?

An electric field is created by any object that has an electric charge. The strength of the electric field is determined by the magnitude of the charge and the distance from the object.

3. Does equal electric field always imply equal potential?

No, equal electric field does not always imply equal potential. While electric field and potential are related, they are different physical quantities and can have different values at the same point in space.

4. What is the relationship between electric field and potential?

The electric field at a point is the negative gradient of the potential at that point. In other words, the electric field is the rate of change of potential with respect to distance.

5. Can an electric field exist without a potential?

No, an electric field cannot exist without a potential. The potential is the measure of the work that must be done to bring a unit charge from infinity to a specific point in the electric field.

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