# Confusion regarding the direction of an electric field

I know this question's been asked extensively on the site. I've read a couple answers here and there, but none of them satisfy my curiosity and confusion.

Here's what I do understand:

• an electric field is an alteration of space caused by a charge

• the alteration of space is the agent that produces a force on nearby charges

Here's my confusion:

Suppose we have a negative source charge and a negative probe charge. We know the two will repel each other, so the force vector for the probe charge points outward, away from the source charge.

My textbook says that if the probe charge is negative (which it is in this case), the electric field vector points in a direction opposite to that of the force vector.

This suggests that the electric field vector points inward towards to the negative source charge, whereas the force vector on the probe charge points outward away from the source charge.

But if the electric field is what generates the force on the probe charge, how exactly is it possible for the inward-pointing electric field to push the probe charge in the opposite direction?

Consider the electric force (Coulomb force) between two charges $q_1$ and $q_2$. It is usually written as $$\textbf{F}_{12} = \frac{1}{4\pi \epsilon_0} \; \frac{q_1 q_2}{r^2}$$ however, an alternative form is \begin{align} \textbf{F}_{12} &= q_1 \cdot \textbf{E}_2 \\ \textrm{ where } \textbf{E}_2 &= \frac{1}{4\pi \epsilon_0} \; \frac{q_2}{r^2} \end{align} is the electric fields generated by the second charge. This second equation is what we are interested in. I will drop the indices and write it as $\textbf{F} = q \textbf{E}$.

• If $q>0$ the force $\textbf{F}$ is parallel to the electric field $\textbf{E}$.
• If $q<0$ the force $\textbf{F}$ is anti-parallel to the electric field $\textbf{E}$.

Therefore, as Philip wrote, there is no logical contradiction involved: The force and the electric field don't have to point in the same direction.

An other example where this is the case is the magnetic force on a moving charge (the so called Lorentz force). Here the magnetic fields is perpendicular to the force.

• Thanks! But in that case, what explains the fact that two electrons repel each other (i.e. that the negative probe charge experiences a force directed outward) if it's not the direction of the field? – AleksandrH Sep 5 '17 at 13:50
• As the formula states, it's not the direction of the E-field alone! It's the interplay between the sign of the charge $q$ and the direction of the field $E$. – Semoi Sep 6 '17 at 6:05

The same field affects positive (probe) charges differently from negative charges. Indeed this is largely what distinguishes positive and negative charges. I can't give you a profound explanation for this. Instead I'll just remark that there's no logical contradiction involved: the same chemical dumped in the water may attract herring but repel haddock.

• Not quite sure I understood your explanation. – AleksandrH Sep 4 '17 at 18:14
• I'd hardly call it an explanation. I addressed just one aspect of what I took to be your question. Can you pin down what was unclear? – Philip Wood Sep 5 '17 at 9:30