Paradox in Induced Dipole Forces: Does it violate Newton's Third Law?

[EM-learner]

Homework Statement

Consider a molecule B with a permanent dipole moment $\vec{p}_B = p_B \hat{z}$ and a polarizable molecule A with atomic polarizability $\alpha$. Molecule A is located on the $z$-axis at a distance $z$ from B.
B creates an electric field that induces a dipole moment $\vec{p}_A$ in A.
Calculate the force $\vec{F}_A$ exerted on the induced dipole A.
Calculate the force $\vec{F}_B$ exerted on the permanent dipole B by the field of A.
Compare $\vec{F}_A$ and $\vec{F}_B$. My current derivation shows $\vec{F}_B = 2\vec{F}_A$, which seems to violate Newton's Third Law.

Relevant Equations

Dipole electric field (on-axis): $\vec{E} = \frac{p}{2\pi\epsilon_0 z^3} \hat{z}$
nduced dipole: $\vec{p}_A = \alpha \vec{E}_B$
Force on a dipole in a non-uniform field: $\vec{F} = (\vec{p} \cdot \nabla) \vec{E}$

First, I calculated the field produced by B at the position of A:

 

[renormalize]

In Physics Forums you should use ## to wrap in-line LaTeX equations and $$ to wrap display equations. I've reformatted your post below:

Homework Statement: Consider a molecule B with a permanent dipole moment $\vec{p}_B = p_B \hat{z}$ and a polarizable molecule A with atomic polarizability $\alpha$. Molecule A is located on the $z$-axis at a distance $z$ from B.
B creates an electric field that induces a dipole moment $\vec{p}_A$ in A.
Calculate the force $\vec{F}_A$ exerted on the induced dipole A.
Calculate the force $\vec{F}_B$ exerted on the permanent dipole B by the field of A.
Compare $\vec{F}_A$ and $\vec{F}_B$. My current derivation shows $\vec{F}_B = 2\vec{F}_A$, which seems to violate Newton's Third Law.

Relevant Equations: Dipole electric field (on-axis): $\vec{E} = \frac{p}{2\pi\epsilon_0 z^3} \hat{z}$
Induced dipole: $\vec{p}_A = \alpha \vec{E}_B$
Force on a dipole in a non-uniform field: $\vec{F} = (\vec{p} \cdot \nabla) \vec{E}$
First, I calculated the field produced by B at the position of A:

And to make our lives easier, please try using LaTeX to post your solution attempt here, rather than making us all open your PDF attachment.

 

[Vincf]

Your mistake comes from differentiating the dipole moment as well, whereas it must be kept constant during the differentiation. To clearly expose the issue, I will redo the calculation in the simplest possible case: a field directed only along the x-axis (which actually corresponds to your situation).

Consider a dipole formed by a charge $q$ at position $x+a$ and a charge $-q$ at position $x$. The force acting on the dipole is:
$$
F = q\,E(x+a) - q\,E(x)
$$

Using a first-order expansion, this gives:
$$
F = q\,a\,\frac{dE}{dx} = p\,\frac{dE}{dx}
$$

This result remains valid even if $a$ is a function of $x$. Physically, this means that the dipole, in its instantaneous configuration, responds to the spatial variation of the field.

In your case, $a$ is a function of $x$, and you write:
$$
q\,\frac{d(aE)}{dx}
$$
which is incorrect.

 

[EM-learner]

In Physics Forums you should use ## to wrap in-line LaTeX equations and $$ to wrap display equations. I've reformatted your post below:


And to make our lives easier, please try using LaTeX to post your solution attempt here, rather than making us all open your PDF attachment.

Thanks a lot !!