Electromagnetism – Can Alike Charges Attract?

Question

Let two positive charges $q_1$ and $q_2$ be travelling in the same direction with the same velocity, say $v$, along $+y$ axis.
The expression for field at a position $\vec{r}$w.r.t charge $q$ moving with a velocity $v$ is $\frac{\mu_0}{4\pi} \frac{q\vec{v} × \vec{r}}{r^3}$.
So $q_2$ would experience a force of $q_2.[\vec{v}×\frac{\mu_0}{4\pi} \frac{q_1\vec{v} × \vec{r}}{r^3}]$= $\frac{\mu_0q_1q_2}{4\pi r^3}[\vec{v}×(\vec{v}×\vec{r})]$
The vector triple product evaluates to $\vec{v}(\vec{v}.\vec{r})-\vec{r}(\vec{v}.\vec{v}) = -v^2\vec{r}. $
The math here tells us that the force would be attractive if the charges are alike! Did I do something wrong? How did this happen?

Answer 1

You have calculated the magnetic contribution to the force in the case where $\vec{v}$ is perpendicular to $\vec{r}$ (incidentally this is not the case for the configuration you described, but it can occur). After you include the electric contribution you will find the force is repulsive overall. However it is interesting to note that the magnetic and electric contributions can be in opposite directions. When the speeds involved tend to the speed of light one can find examples where the net force tends to zero, but it never changes sign.