# When does electrical charges cancel each other?

1. If we have a wire with equal amount of positive and negatvie charges, will the total charge of the wire then be neutral?
2. If the first statement is true. Then a positiv charge will cancel with a negative charge. Does this mean that a positive charge can only "focus/attract" one negative charge at a time? For example if we had three isolated charges. One positive, and two negative. Will the positive charge then attract one of the negative charges. And the last negative charge will stay stationary and will not feel any attraction?
3. If the second statement is true. Can we then put a positive charge outside a wire, and let one of the negative charges inside the wire focus on the charge outside. Will the wire then be positive?

• Yes, there is an equal amount of electrons -negative charges- and protons -positively charged. Therefore the net charge, which is the sum of all charges, will be zero.
• No! In the example of two negative charges and one postive charge. The postive charge will attract both negative charges and the force on the positive charge will be the vectorsum of the force exerted by negative charge 1 and negative charge 2. This force can be easily calculated by Coulomb's law: $$\vec{F}=\frac{q_1q_2\vec{r}}{4\pi\varepsilon_0r^3}$$

To complement QuirkyTurtle89's answer, in any system, a charged particle will feel interactions from all the other charged particles, positive or negative.

The total charge of a system or subsytem is sometimes a useful concept in physics. But if you want to talk about particles attraction/repulsion, what you need to calculate is the force that each particle feels from its interaction with the others. Let's see what the difference is by using a three particles example:

Assume the particles at positions $\vec{r_1}$, $\vec{r_1}$ and $\vec{r_3}$, with respective charges $q_1 e$, $q_2 e$ and $q_2 e$.

The force that the particle 1 feels from the others is the sum of the Coulomb forces with each of the other particles:

$$\vec{F_1} = \frac{q_1 q_2}{4 \pi \epsilon_0}\frac{\vec{r_2} - \vec{r_1}}{||\vec{r_2} - \vec{r_1}||^3} + \frac{q_1 q_3}{4 \pi \epsilon_0}\frac{\vec{r_3} - \vec{r_1}}{||\vec{r_2} - \vec{r_1}||^3}$$

Let's assume particles 1 and 2 have a charge $-e$ and particle 3 has $+e$. The force on particle 1 becomes

$$\vec{F_1} = \frac{1}{4 \pi \epsilon_0} \left( \frac{\vec{r_2} - \vec{r_1}}{||\vec{r_2} - \vec{r_1}||^3} - \frac{\vec{r_3} - \vec{r_1}}{||\vec{r_2} - \vec{r_1}||^3} \right)$$

Here you see the funny thing: Particles 2 and 3 have opposite charges, so the total charge of the two particles is zero. But by summing forces from particles 2 and 3 rather than charges you obtain a force feeled by the first particle that is generally not zero. In other words, to talk about attraction and repulsion of charges, you need to consider the charges between the particles but also their positions, as the further away particles are, the lower the interaction.

If you want to look further into this three-particle system, I suggest you read about electrostatic dipoles, as they are a typical example of a system of two particles with not net charges.