Fill a glass with water, raise it up high in the air, and invert it. You see the water spill down to the ground. If any naive buddy happened to ask you the reason for this downward falling of water from the glass, you’d right away fire back that it was obviously the gravity that was “pulling” the water downwards. You might even caste a look of contempt at them for their naiveness! But say, this buddy of yours was only pretending to be dumb, and now asks a serious question as for what exactly causes the electricity in the electric wires to flow. Most of us would still find the question naive and yet not have a decent answer to it. Electricity flows in the wire because it is meant to flow so! Now, this is the buddy’s turn to return the frown. An electric field inside the wire is what causes the electric current to flow through the wire. Haven’t you ever studied the Ohm’s law?
Before you might possibly get into such an embarrassment for real, it’s best to catch the concept well and clear. The situation is not very different from the water spilling off a glass due to gravity. Just like a gravitational field is responsible for the flow of water, an electric field is what causes an electric current to flow, in general. Let’s see how better we can understand the situation and get used to looking at the electric current flow as intuitively simple as the spilling water, with simple examples.
To begin with, let’s straightaway look at the conventional way of defining the electric current. Electric current is generally defined as the rate of flow of electric charges past a given point. That is, the electric current at any specific point in space, is a measure of how quickly a certain amount of charge flows past that point. That “certain amount of charge” might simply be envisaged as a cluster of charges (such as the electrons or the protons, the most popular of the charged particles). Mathematically, if is the quantity of charge we’re concerned with, the electric current is . Where, is the time taken by the charge to pass through the point of measurement. Note that here we’re restricting ourselves to the simple case of a steady current flow – a flow in which the charges move past a point in space, as quickly at any particular time as they do at any other time during the flow.
Now, if the electric current were to be confined to flow through a cylindrical tube, say (like water flowing through a hose pipe); we can define a useful quantity called the electric current density as the amount of current flowing per unit cross sectional area of the tube,
That is, while the electric current tells us how much charge passes a point in a time interval, the electric current density tells us how much current has passed through a surface (of area ) in that time interval. You can imagine this surface to be an imaginary mesh in the path of the flow (as you can see in the image below) that is placed “perpendicular” to the flow direction. We’ve also assumed here that the same number of charges cross any part of the surface as that that cross any other part (of the same area) of the surface – there’s an even flow of charges through every part of the mesh.
So far, a lot of assumptions and simplifications have pilled up. For example, you might ask, why should the mesh be placed only “perpendicular” to the flow direction? Why should there be an even flow of charges? And so on. Let us now break free of all these assumptions at once and let us see the true beauty of the concept unfold in all its complexity. Consider an uneven flow of current in which different number of charges cross through different parts of the mesh in a specific interval of time. In fact, let us even rid off the plane mesh, and introduce an uneven surface, like a crumpled paper straightened out. What is the total current that passes through the surface in any specific interval of time? If it weren’t for the complexities we’ve just assumed, we could simply have used equation (1) and found out the current to be . But here, is different at different parts of the surface area (uneven flow); so there’s no single value of we can simply plug into the equation. To solve the problem, we cleverly hypothesise that might not be the same all over the area , but if we choose only a tiny enough part of , the current density through that speck of surface can be assumed to be more or less constant (just like the earth just beneath your house is more or less flat, because your house occupies a tiny speck of the earth’s spherical surface). So, we chop the surface (in our mind) into a large number of chunks; large enough so that each chunk would be so small, that the current density over it at least, would be approximately constant. Let’s call this tiny chunk of surface area , and the tiny current passing trough it, . Note that if you add up all the , you get back the total area of the surface ().
What’s the total electric current?
By now, you might be confident that the current through the surface can be found by considering one chunk at a time, and writing the tiny current through each chunk as , and then adding up all the tiny contributions, to get the total current through the entire surface. But in doing so, we would be overseeing the fact that the orientation of each of the chunks is very important in deciding how much current passes though it. For example, since we’ve considered a surface that looks like a crumbled paper, there can be parts (chunks) of the paper that are “parallel” to the flow direction, and so no current at all flows through the chunk. The orientation of the chunks therefore has to be accounted, and we do so by writing the differential current as a dot product of the current density and area vectors,
The direction of the current density vector being that of the current flow. (The vector area is defined as goes the norm, it tells us the value of the area it represents and points in a direction perpendicular to the surface, pointing either way.) For a more thorough explanation on the effect of the surface’s orientation, read this (under “The flux”).
And so, the total current through the entire surface obtains by adding up all the tiny contributions (i.e., by integrating),
From the above equation, all those assumptions we’d started with, become apparent. If the area is always “perpendicular” to the flow direction, . If the flow is even, is constant, and so, – back to where we began!
Note that Equation (3) tells us of a more general situation than equation (1) does. We can say that (1) is obtained from (3) under special conditions (even flow, perpendicular surface, etc), as we’ve demonstrated.
So far, we’ve been speaking of charges in motion and their effects, without worrying of what exactly got the charges moving in the first place. As we’ve seen earlier, electric charges react promptly to electric fields. Therefore, the simplest way to get an electric current flowing is to apply an electric field in the region of interest. The positive charges all begin moving in the direction of the field, while the negative ones move in the opposite direction. And by convention, we choose the direction of motion of the positive charges as the direction of the electric current. Compare this with the action of the gravitational field on material objects. Just like gravity causes masses to plummet, electric fields cause positive charges to accelerate.
We’ll continue from here in the next article, where we’ll go on to make a very profound conclusion – the Ohm’s law; and discuss the law’s further implications.
Cover image by @wewon31