# Curl of an electric dipole field

• Identity
In summary, the conversation discusses the topic of whether the field of an electric dipole is conservative. The speaker initially thought it would be, but after researching vector calculus, they discovered that the curl of the gradient of a function is equal to zero. This leads to a discussion about a positively charged rod in an electric dipole field and whether it rotates, which suggests that the field may not be conservative. The speaker also mentions that this applies to electric monopole fields as well.
Identity
Is the field of an electric dipole conservative?

Initially I thought it would be, for no particular reason but that's just what my high school intuition thought. (haha I thought everything would be conservative apart from friction)

But I was reading up on some vector calculus and discovered

$$\mbox{curl}(\nabla f) = \mathbf{0}$$

If you put a positively charged rod in an electric dipole field, and fix it at the right orientation, it will rotate. Does that mean that the field is not conservative?

EDIT: I just realized it would also rotate in an electric monopole field lol

thanks

It just means you have to include rotational as well as translational energy in applying conservation of energy.

for pointing that out!

it is important to clarify that the statement "everything is conservative except for friction" is not entirely accurate. While it is true that many physical systems can be described by conservative forces, there are also cases where non-conservative forces, such as friction or drag, play a role. Therefore, it is important to always carefully consider the specific system and forces involved before making assumptions about their conservativeness.

Regarding the question about the curl of an electric dipole field, it is important to first clarify what is meant by "field of an electric dipole." An electric dipole is a pair of equal and opposite charges that are separated by a small distance. The field of an electric dipole is the electric field that is produced by these charges.

In this context, the curl of an electric dipole field is not necessarily zero. The curl of a vector field is a measure of its vorticity, or the tendency for the field to rotate around a point. In the case of an electric dipole field, the electric field lines do not form closed loops, which means there is a non-zero curl. This is in contrast to an electric monopole field, where the electric field lines do form closed loops and the curl is zero.

In terms of conservativeness, a conservative force is one that can be described by a potential function. The curl of a conservative force is always zero. Therefore, based on the fact that the curl of an electric dipole field is non-zero, we can conclude that the field of an electric dipole is not conservative. This is because the electric field is not a conservative force, but rather a non-conservative force that can produce a torque on an object with a dipole moment, causing it to rotate.

In summary, the field of an electric dipole is not conservative due to the non-zero curl of the electric field. This is an important concept to understand in the study of electromagnetism and can have implications for understanding the behavior of objects in electric fields. It is always important to carefully consider the specific system and forces involved before making assumptions about their conservativeness.

## 1. What is the definition of the curl of an electric dipole field?

The curl of an electric dipole field is a measure of how much the field is rotating or circulating around a given point. It is a vector quantity that describes the direction and magnitude of the rotational component of the electric field.

## 2. How is the curl of an electric dipole field calculated?

The curl of an electric dipole field is calculated using a mathematical operation known as the vector cross product. This involves taking the derivative of the electric field with respect to each coordinate axis and then combining these derivatives using specific rules to determine the magnitude and direction of the curl.

## 3. What does the curl of an electric dipole field tell us about the field itself?

The curl of an electric dipole field provides insight into the behavior of the field near a given point. A non-zero curl indicates that the field is rotating or circulating around that point, while a zero curl suggests that the field is not changing in a rotational manner.

## 4. How is the curl of an electric dipole field related to electromagnetic waves?

The curl of an electric dipole field plays a key role in the generation and propagation of electromagnetic waves. It is a crucial component of Maxwell's equations, which describe the behavior of electromagnetic fields and how they interact with each other.

## 5. Can the curl of an electric dipole field be measured experimentally?

Yes, the curl of an electric dipole field can be measured experimentally using various techniques such as Faraday rotation or optical interferometry. These methods allow for the detection and measurement of changes in the electric field and can be used to determine the curl at a given point.

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