How should I determine the direction of the force in the given case: (p is the electric dipole moment)
Using the following formula
My book has the following :
I am unable to understand the given correlation between the direction of Force and the formula. I am confused with the gradient of E and it's dot product (correct me if I am wrong) with p .
take advantage of the fact that $\vec p$ is a vector.
Resolve $\vec p$ into two components - $\vec p_1$ and $\vec p_2$ one along the line joining charge $q$ and the centre of dipole and one perpendicular to it (ref fig 1). the
Electric field at an axial point of dipole : $\vec E_{axial}= \frac{1}{4\pi\epsilon_0}\frac{2p_2}{r^3}$
Electric Field along the equitorial plane : $\vec E_{eq}= \frac{1}{4\pi\epsilon_0}\frac{p_1}{r^3}$
Now, the net electric field at $q$ is $\vec E=\vec E_{eq}+\vec E_{axial}$ and force on $q$ is just $q\vec E$
Think of a “finite-size dipole” as a positive charge at the tip of the $\mathbf p$ vector and an equal negative charge at the tail. The forces exerted by $q$ on these two charges will not be exactly equal and opposite, because the field of $q$ isn’t uniform and the two charges aren’t at the same point. Draw the two force vectors and add them to get the net force on the dipole.
