In introductory physics textbooks often have some version of the following two problems.
- Here are two electric charges. Where is the electric field equal to zero (technically the zero vector)?
- Here are two electric charges. Where is the electric potential (with respect to infinity) equal to zero?
Yes, the electric field is a vector and the electric potential is a scalar — so you would think that the question about potential might be simpler, but not so. If you only have two electric charges, the electric field vector can only be zero on an axis connecting the two charges.
Here is my explanation of the location of the zero electric field.
But enough about the electric field. The fun part is the electric potential. It’s actually sort of complicated and at this point, I don’t even know all the answers. Let’s figure this stuff out.
Here is the situation. Two charges, q_1 and q_2. I’m going to put one of them at the origin and the other on the x-axis. Like this.
I will start with the simple solution — it’s the answer to the question “where on the x-axis is the electric potential equal to zero?” The electric potential at any point is the sum of the potential due to each point charge. Yes, I’m assuming that this is the potential with respect to infinity — I just have to get that out of the way. But here is the potential for a point charge.
In this expression the 1/(4πε_0) is a constant (also represented by just “k”). Of course “q” is the charge and r is the magnitude of the vector distance from the charge to the point where you want to find the potential. So, for two charges, I need two vector positions. Let’s do this the vector-way from the beginning. The two charges will be at the vector locations r_q1 and r_q2. With that, I can find the vector from each charge to the observation location (where I want to find the potential). I’ll call the observation location r_obs (sorry about the lack of subscripts — not my fault).
Once I find the magnitudes of these two vectors, the total potential would be:
OK, but we are stuck on the x-axis (for now). Let’s put the observation location at some value — x. That means r_1 is equal to x (since it’s at the origin) and r_2 is equal to x-s. The potential can then be written as:
Notice that I need to take the absolute value of x since the potential depends on the magnitude of the vector r. But this means that the two charges (q_1 and q_2) must have opposite signs. There’s no way they can add up to zero if they are both positive (or both negative). So, let’s say q_1 is positive and q_2 is negative.
Now let’s consider the region to the right of q_2 (labeled region C). If I pick a point where x is greater than s, then the only way for V to be zero is if q_2 is a smaller magnitude than q_1. This is because q_2 will be divided by a smaller number (since x-s is always smaller than x in this region). Well, let’s go ahead and find the value of x. I can drop the absolute value signs because x is already positive. Also, the k value will cancel so that the following must be true:
Solving for x:
Suppose I have a charge q_1 = 2 nC and q_2 = -1 nC with s = 0.1 meters. Putting in these values, I get x = 0.2 meters for a zero potential.
Great. But what about region B (in between the two charges). I can write the total potential as:
Solving for x (I’ll skip all the steps — it’s similar, but different), I get:
Remember, this solution assumes that q_1 and q_2 have opposite charges. But if I use the same values as above, then I get another zero potential at x = 0.0667 meters. This is closer to charge q_2 since it has a lower value of charge. That makes sense.
For the last region (A), there isn’t a location for a zero potential. If q_1 is greater than q_2 then the potential due to q_1 will ALWAYS be greater in this region since that charge is closer to every x value.
OK, I think you can really see everything with a plot. This graph shows the potential due to both charges along with the total potential. Since the potential due to a single point blows up, I cut that put off. Here’s what I get. the blue line is the total potential.
I put two dots at the locations where V = 0 volts.
All the Other Points
But what about locations that aren’t on the x-axis? There are still more points to find since the potential is a scalar quantity — as long as the two potential values have equal and opposite values, they can cancel. Let’s start with region B (in between the charges). Here is a diagram.
The potential will be zero if the following is true (the k’s cancel).
Squaring both sides and cross multiplying (I hate saying that, but I want to skip some steps):
Just to simplify stuff, I’m going to divide both sides by q_1² and then make a new constant:
I mean, it’s no big deal. It’s just a constant. But that make it easier to get an expression for y²:
Here is part of a plot showing some of these zero potential points (I have also put a point where the two charges are).
OK, that’s it. But wait! I have another idea to find these zero potential points — I’ll save that for a later post.