What is the force on an elementary dipole from a point charge in the same plane?

In summary, the conversation discusses how to show that the force on an elementary dipole of moment ##\mathbf{p}##, distance ##\mathbf{r}## from a point charge ##q## has components along and perpendicular to ##\mathbf{r}## in the plane of ##\mathbf{p}## and ##\mathbf{r}##. The relevant equations include the potential, force, and dipole moment. It is noted that the force is the gradient of the potential and that choosing p to be along the z-axis simplifies calculations.
  • #1
andre220
75
1

Homework Statement


Show that the force on an elementary dipole of moment ##\mathbf{p}##, distance ##\mathbf{r}## from a point charge ##q## has components
$$\begin{eqnarray}
F_r &=& -\frac{qp\cos{\theta}}{2\pi\epsilon_0 r^3}\\
F_\theta &=& -\frac{qp\sin{\theta}}{4\pi\epsilon_0 r^3}
\end{eqnarray}$$
along and perpindicular to ##\mathbf{r}## in the plane of ##\mathbf{p}## and ##\mathbf{r}##, where ##\theta## is the angle which ##\mathbf{p}## makes with ##\mathbf{r}##.

Homework Equations


$$\Phi(\vec{r}) = \frac{1}{4\pi\epsilon_0}\frac{\vec{p}\cdot\vec{r}}{r^3}$$
$$F = -\frac{d\Phi(\vec{r})}{dr}$$
$$\vec{p} = Q\vec{r}$$

The Attempt at a Solution


Frankly, I do not know how to start this one. I need to find the force on the charge from the dipole. To do so, I take the derivative of ##\Phi## and find the force using Coulomb's law equation. And them decompose it in terms of ##r## and ##\theta##. Is this the right direction?
 
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  • #2
Note that the force is the gradient of the potential. What you have written down in the relevant equations is simply the r-component.

Also, please show us what you get when you try to do what you said.

Note: ##Q\vec r## is the dipole moment of a point charge ##Q## in ##\vec r## with respect to the origin. This is not the situation in your problem, which has a fundamental dipole ##\vec p##.
 
  • #3
andre220 said:
$$F = -\frac{d\Phi(\vec{r})}{dr}$$

Hi. The above equation is not right and you can tell because force is a vector and your expression gives you a scalar.
In 3 dimensions: F = –∇Φ, which is a gradient and you'll have to look up how to take it in spherical coordinates because the expression is far from obvious.
Lastly, without loss of generality you can take p to be along the z-axis so that pr = pr cosθ, which should simplify your calculations...
 
  • #4
Sorry, didn't see Orodruin's post... I'll leave it to him from now on.
 
  • #5
Yes, thank you. It was quite simple once you pointed out the fact that ##F## should be a vector. And that ##p\cdot r = p r \cos{\theta}##. Thank you for your help.
 
  • #6
No problem. I just have to make it clear that although you have in general p⋅r = pr cosδ, where δ is the angle between p and r, you can take δ = θ (the polar angle of your spherical coordinates) only if you have freedom in choosing the direction of p, which is the case here. It's a detail but always important to understand clearly...
 

Related to What is the force on an elementary dipole from a point charge in the same plane?

1. What is an elementary dipole?

An elementary dipole is a fundamental unit in physics that represents two equal and opposite charges separated by a small distance. It is used to describe the distribution of charges in a system.

2. What is the force on an elementary dipole?

The force on an elementary dipole is the result of the interaction between the two charges that make up the dipole and an external electric field. It can be calculated using the equation F = qE, where F is the force, q is the charge, and E is the electric field.

3. How does the orientation of an elementary dipole affect the force on it?

The orientation of an elementary dipole can affect the force on it. When the dipole is aligned with the electric field, the force is maximum. When the dipole is perpendicular to the electric field, the force is zero. When the dipole is anti-parallel to the electric field, the force is minimum.

4. What is the relationship between the force on an elementary dipole and the distance between the charges?

The force on an elementary dipole is directly proportional to the distance between the charges. As the distance between the charges increases, the force on the dipole decreases. This can be seen in the equation F = qE, where the force is inversely proportional to the distance between the charges.

5. Can an elementary dipole experience a torque in addition to a force?

Yes, an elementary dipole can experience a torque in addition to a force. This is because the two charges that make up the dipole are separated by a distance, creating a moment arm. When the dipole is placed in an external electric field, the force acting on each charge creates a torque on the dipole, causing it to rotate.

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