Understanding the Coordinate-Free Electric Field of a Dipole

In summary, the conversation discusses the concept of a "coordinate-free" electric field of a dipole and its representation in terms of unit vectors. The purpose of this is to understand the meaning of the question and how it relates to the given equations. The equation for the electric field of a "pure" dipole is also provided.
  • #1
raddian
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I am "continuing this thread" in hopes of asking questions that deal with the meaning of the question. https://www.physicsforums.com/threa...dipole-moment-in-coordinate-free-form.359973/
1. Homework Statement

Show that the electric field of a "pure" dipole can be written in the coordinate-free form
$$E_{dip}(r)=\frac{1}{4\pi\epsilon_0}\frac{1}{r^3}[3(\vec p\cdot \hat r)\hat r-\vec p].$$

Homework Equations


$$E_{dip}(r)=\frac{p}{4\pi\epsilon_0r^3}(2\cos \hat r+\sin\theta \hat \theta)$$

The Attempt at a Solution


I am trying to understand what "coordinate free" means. If the answer is in terms of r hat and theta hat, doesn't that contradict "coordinate free"? AND i would get $$ p = pcos(\theta) \hat r - psin(\theta) \hat \theta $$. Why doesn't p depend on PHI? If it's coordinate free why are we restricting our coordinates to r and theta??
 
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  • #2
"Coordinate free" means you don't need to define the coordinate system to write your equation. ##\hat{r}## is a unit vector from the center of the dipole to the observation point, so given the orientation of ##\mathbf{p}## in space, the relative direction of ##\hat{r}## with respect to ##\mathbf{p}## will automatically follow.
 

1. What is a dipole?

A dipole is a pair of equal and opposite electric charges separated by a small distance. This creates an electric dipole moment, which is a measure of the strength and direction of the dipole.

2. How is the electric field of a dipole calculated?

The electric field of a dipole can be calculated using the formula E = (1/4πε0) * (p/r3) * (3cosθ * r1 - r), where ε0 is the permittivity of free space, p is the electric dipole moment, r is the distance from the dipole, and θ is the angle between the dipole axis and the direction of the field.

3. What does it mean for the electric field of a dipole to be coordinate-free?

A coordinate-free electric field of a dipole means that the field can be described without using a specific coordinate system. This is important because it allows for a more general understanding of the field, as it is not dependent on any specific orientation or reference frame.

4. How does the direction of the electric field change for different points around a dipole?

The direction of the electric field changes depending on the distance from the dipole and the angle at which it is measured. At points close to the dipole, the field lines are directed towards the positive charge, while at points further away, the field lines become more parallel to each other.

5. What are some real-world applications of understanding the coordinate-free electric field of a dipole?

Understanding the coordinate-free electric field of a dipole is crucial in many areas of science and technology. It is used in the design of electronic circuits, in medical imaging techniques such as MRI, and in the study of molecular structures. It is also important in understanding the behavior of polar molecules in electric fields.

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