# How will the current flow in the figure drawn below?

Consider the green material in the figure to be conducting. So, I was wondering how the current will flow in the rod, as the battery is not connected at the ends, but on the surface of the rod. Thus, will there be no current at the ends of the rod and what about voltage?

An approximate numerically calculated figure is attached. The figure is a two-dimensional result. Although 3D calculations is possible, the figure is easier to see in 2D. $$\phi$$ is potential in volts. $$\vec{J}$$ is the electric current density.

(Edit #1) I add a figure highlighting the elements in the two left-most columns.

• Excellent answer! It is interesting how far the current goes. There is definitely current further away from the contact point than I thought there would be
– Dale
Jun 10 at 17:42
• This trips up people with ground/power plane design too. People tend to be surprised by how much a heavy load can affect voltages elsewhere on the plane, even on points 'the other side' of the main connection. It would be interesting to redo this with deep slots (cut away the 6th-and-10th set from the left of rectangles, except for the top one) cut from the bottom.
– TLW
Jun 11 at 0:03
• Out of curiosity, how did you simulate this? Jun 11 at 14:33
• @TLW, the "cut from the bottom" calculation itself is simple and easy, but it takes time and effort to write a report that accurately captures the new results. That is why it cannot be done here. Jun 11 at 22:42
• @Landak, I performed this calculation with a finite element (FE) code of my own development and plotting is performed by the GiD pre/post processor. My code is still incomplete; having bugs and documents is in short. Of the many types of electromagnetic FE calculations, this calculation is one of the easier ones, solving Laplace's equation with fixed potentials at two points. For example, if you have access to a finite element code that can solve temperature calculations, you should be able to perform similar calculation with that code. Jun 11 at 22:43

Yes. There won't be any noticeable amount current at the ends of the rod because the electric field formed inside the conductor will be restricted so the electrons will dominantly experience the force when they are in the region between the points across which the potential difference is applied or else there might be some small amount influence of electric field at the end and the area near the end, which will result in some negligible amount of current.

And the same thing goes with potential difference. The potential at one end and the point at which the wire is connected will be nearly same and hence the potential difference will be nearly equal to zero and you won't find any significant amount of current behind those two points at which the wires are connected.

While doing some ideal calculations you can also neglect the resistance of that region of the rod which is on the left of positive terminal and same with which is on the right of negative terminal to get some near to accurate results.

• I don't understand your geometry. "that region only across which the wires are connected" Do you mean no current to the left of the conductor on the left? "any current behind those two points" do you mean, for example, to the left of the conductor on the left? Jun 10 at 13:45
• @garyp yes that's what I mean. In other words it can be said that the electrons will flow in the region across which the potential difference is applied by the battery. Except that region, you will approximately find that there is no current. Jun 10 at 14:19
• Ok, I wanted to clarify that because with that interpretation your answer is not correct as the other answers here show. Jun 11 at 3:18
• Look at the simulation result in the other answer. It shows that there is current at the end of the rod. Jun 11 at 5:14
• The edit improves the answer, but I still object to a few things. Whether or not the current is noticeible depends the sensitivity of the detector. The word negligible.: negligible is in the eye of the beholder. That is, whether or not the current is negligible depends on what you want to know about the system. The same applies to whether or not you neglect the resistance in the regions "outside" of the wires. Jun 11 at 15:20