Force on a charge from an induced dipole

In summary, the force of attraction between a point charge q and a neutral atom with polarizability α at a large distance r is given by: F = -αq^2 / 8π^2ε0r^5 in the direction of the negative unit vector r. This may seem unexpected because the force points towards the origin, but it is correct based on the equations for a monopole and dipole.
  • #1
Yitzach
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Homework Statement


A point charge q is situated a large distance r from a neutral atom of polarizability [tex]\alpha[/tex]. Find the force of attraction between them.

Homework Equations


[tex]\vec{E}_{mono}(r)=\frac{q}{4\pi\epsilon_0r^2}\hat{r}[/tex]
[tex]\vec{E}_{dip}(r,\theta)=\frac{p}{4\pi\epsilon_0r^3}(2\cos\theta\hat{r}+\sin\theta\hat{\theta})[/tex]
[tex]\vec{p}=\alpha\vec{E}[/tex]

The Attempt at a Solution


[tex]\vec{E}_{mono}(r)=\frac{q}{4\pi\epsilon_0r^2}\hat{r}[/tex]
[tex]\vec{p}=\alpha\vec{E}[/tex]
[tex]\vec{p}=\frac{\alpha q}{4\pi\epsilon_0r^2}\hat{r}[/tex]
[tex]\vec{E}_{dip}(r,\theta)=\frac{p}{4\pi\epsilon_0r^3}(2\cos\theta\hat{r}+\sin\theta\hat{\theta})[/tex]
[tex]\vec{E}_{dip}(r,\pi)=\frac{\alpha q}{16\pi^2\epsilon^2_0r^5}(-2\hat{r})[/tex]
[tex]\vec{E}_{dip}(r,\pi)=-\frac{\alpha q}{8\pi^2\epsilon^2_0r^5}\hat{r}[/tex]
[tex]\vec{F}=q\vec{E}[/tex]
[tex]\vec{F}=-\frac{\alpha q^2}{8\pi^2\epsilon^2_0r^5}\hat{r}[/tex]

The result I got was unexpected because that is a repulsive force.
Do I need to go about a longer way or did I mess it up somewhere?
 
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  • #2
Why do you say that the force is not attractive? It points in the negative ##\hat r## direction, i.e. towards the origin where presumably you put the monopole.
 

Related to Force on a charge from an induced dipole

What is "force on a charge from an induced dipole"?

"Force on a charge from an induced dipole" refers to the force experienced by a charged particle when it interacts with an electric dipole that has been induced in a nearby object.

What causes an induced dipole?

An induced dipole is caused by the presence of an electric field. When an electric field is applied to a neutral object, the charges within the object will redistribute, resulting in a separation of positive and negative charges and the creation of an induced dipole.

How does an induced dipole affect the force on a charged particle?

An induced dipole creates a non-uniform electric field, which exerts a force on a charged particle within its vicinity. This force can either attract or repel the charged particle, depending on the orientation of the dipole and the charge of the particle.

Can the strength of the force on a charged particle from an induced dipole be calculated?

Yes, the strength of the force can be calculated using the equation F = qE, where F is the force, q is the charge of the particle, and E is the electric field strength of the induced dipole.

What are some real-world examples of the force on a charged particle from an induced dipole?

Some examples include the way water molecules interact with charged ions in a solution, the way charged particles are attracted to a charged balloon, and the way dust particles become electrically charged when rubbed against a surface.

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