Can an ideal dipole experience an electric force? It is known that electric force a charged body is given as $\vec{F} = q \vec{E}$ given that $\vec{E}$ is uniform. Now, for an ideal dipole, what would we take as the charge for calculating the force exerted on it by an external electric field?
 A: The electric force on a dipole is $\mathbf F=(p \cdot \boldsymbol \nabla ) \mathbf E$, where $p$ is the dipole moment, $\mathbf E$ is the electric field, and $\boldsymbol \nabla$ is the gradient operator.
Note that this force goes to zero for a uniform field (as mentioned in another answer). Note also that there will be a torque to align the dipole with the field.
A: You can simply use the formula you stated to compute the force on each component of the dipole separately and then you add them up to get the net force. Note that this will be zero unless the electric field varies in space.
To give more details, consider a dipole of positive charge $q$ with separation $d\,$ lying along the $x$-axis such that the charges are at $x = 0, x = d$.
We will take the ideal limit at the end. The force on the positive charge is $qE(d)$, while that on the negative charge is $-qE(0)$.
The total force is then $$F = q(E(d) - E(0)) = qd\frac{E(d) - E(0)}{d}$$ Now, the ideal dipole limit is $d \to 0$, with $qd = p$ fixed. We are thus left with $F = p\frac{dE}{dx}\Bigl|_{x=0}$.
