Solving for Griffiths torque on a dipole

[ehrenfest]

[SOLVED] Griffiths torque on a dipole

Homework Statement

This question refers to Griffiths E and M book.

In the sentence after equation 4.5, Griffiths says the following:

"For a perfect dipole of infinitesimal length, Eq 4.4 gives the torque about the center of the dipole..."

What is driving me insane is that I thought eqn 4.4 was true precisely when the dipole was finite since they say nothing about d being an infinitesimal there and the vector d sure does not look like an infinitesimal in Figure 4.5? I was sure that eqn 4.4 applied only for physical dipoles before I read that sentence! What is wrong here! Please help.

Homework Equations

The Attempt at a Solution

 

[kdv]

Homework Statement

This question refers to Griffiths E and M book.

In the sentence after equation 4.5, Griffiths says the following:

"For a perfect dipole of infinitesimal length, Eq 4.4 gives the torque about the center of the dipole..."

What is driving me insane is that I thought eqn 4.4 was true precisely when the dipole was finite since they say nothing about d being an infinitesimal there and the vector d sure does not look like an infinitesimal in Figure 4.5? I was sure that eqn 4.4 applied only for physical dipoles before I read that sentence! What is wrong here! Please help.

Homework Equations

The Attempt at a Solution

What he does is to do the calculation for a finite $\vec{d}$ in a uniform E field and then consider the limit as d becomes infinitesimal. There is nothing wrong with doing it this way. The only thing is that for an infinitesimal dipole, the product $q \vec{d}$ is not well-defined. But if you introduce a vector $\vec{p}$ for an infinitesimal dipole, then eq 4.5 is completely fine. So the only subtle point i sthe definition of the vector p for a perfet dipole. If it is defined as the infinitesimal limit of the vector "q d" then the whole procedure is well-defined.