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The electric current and Ohm’s law

Electronics

The electric current and Ohm’s law

Cover image: flickr.com/Allen Levine

The electric current and Ohm’s law

Beginning with an understanding of the meaning of electric current, we arrive at a remarkable result, the Ohm’s law, with the aid of simple assumptions.

The electric current was simply defined earlier as a measure of how quickly electric charges flow through a point. If a large number of charges streamed through a point every second, we said the electric current is large. But taking a step back, we question: what in the first place caused these charges to move? We obviously know that charges move only when they are made to, not when they want to. Let us explore the possible ways in which we can get charges moving, and look at some of its consequences.

Getting the charges to move

All matter is made of charged particles – there are positive nuclei and negative electrons. So, if you’re planning to move charges, just move any material object (really?). Even as you wave your hand through air, the positive and negative charges that make your hand, are indeed moving along. Does that mean you’ve just generated an electric current just with a wave of your hand? Of course, not. The fact is that your hand is normally electrically neutral – it is made of equal number of positive and negative charges. So, the total electric charge on your hand is zero. Hence simply waving your hand through air, in no way generates any electric current.

Therefore, we need a net charge in the first place, to even start thinking of an electric current. Consider the simplistic example of a single positively charged particle. As we know, this single charge cannot constitute an electric current. It simply isn’t possible to measure a “rate of flow of charge through a point per second”, for the charge passes through any given point only once, and the next moment it’s gone out of the picture. But we consider the single charge now, to understand how we can get it moving.

If the charge is initially stationary, we could get it moving by applying a mechanical force. But a more natural way to do this is to pass an electric field. As we’ve seen before, much like massive objects fall in a gravitational field, charged particles accelerate in an electric field. So, even if we have a bunch of charges, the simplest way to get them moving and hence generate an electric current is to pass an electric field through the space they are in.

Mathematically, we say that the electric current density is directly proportional to the electric field applied. 

    \[\mathbf{J} \propto \mathbf{E}\]

 And introduce the proportionality constant \sigma, the “electrical conductivity”.

(1)   \begin{equation*} \mathbf{J}=\sigma \mathbf{E}  \end{equation*}

This equation is true in most cases (we’ll discuss the exceptions at a later time). So, if you want an electric current flowing through a conductor, all you have to do is pass an electric field through it. This field will apply a force (\mathbf{F}=-e\mathbf{E}) on each free electron (of charge -e) present in it, producing an electric current.

How electric current really flows

Now, you might be imagining a smooth flow of electrons, as soon as the electric field is applied – something like water flowing through a hose pipe, steadily. But in reality, the electrons move in a very higgledy-piggledy manner. They keep bumping into the fixed atoms of the material as they make their way along the applied electric field. You can imagine it something like, each electron bumps into an atom, comes to a stand still, and then again accelerates along the field, until it bumps into another atom in its way and again stops momentarily. As if this weird motion wasn’t enough, to add to the complexity, the electrons experience other forces from the atoms, too. To make some sense of the chaos, we disregard these additional forces on the electrons, by saying that the electrons behave just like free particles, albeit with a different mass m^*, instead of its actual mass m. in other words, mentally freeing the electrons from the clutches of the atoms, we are paying with a change in their observed mass.

Applying Newton’s second law of motion, we get the acceleration of each electron to be

    \begin{align*} \mathbf{a}=&{\mathbf{F} \over m*}\\ =&{-e\mathbf{E} \over m*} \end{align*}

Since, in our model, we have assumed that the electron starts afresh after each bump, from rest, the velocity gained in a time t, called the drift velocity, will be,

    \begin{align*} \mathbf{v}=&\mathbf{a}t\\ =&\frac{-e\mathbf{E}}{m*}t \end{align*}

If t=\tau is the mean free time – the average time between collisions, the average velocity gained by the electrons during their motion is,

    \[\mathbf{v}={-e\tau\mathbf{E} \over m*}\]

 We call the quantity \frac{e\tau}{m*}, the mobility \mu, which is a measure of how quickly the electron can move through the material. In other words, it is the drift velocity of an electron when a unit electric field is applied. It depends on the details of the structure of the material.
Therefore,

(2)   \begin{equation*} \mathbf{v}=-\mu \mathbf{E}  \end{equation*}

Calculating the current

All these electrons that are drifting along the conductor, under the applied electric field, constitute an electric current. To find the electric current, we just have to find the quantity of charge (or the number of electrons) that will cross an imaginary surface (placed perpendicular to the flow direction) in one second. Let dA be the area of a tiny chunk of the surface. Your job now is to wait at the surface and carefully count the number of electrons that cross it in one second. Now, either you can go ahead and literally count, or you can use the simple logic that the only electrons that will cross the surface in one second are the ones that are not farther than one-second’s drive from the surface. By that it is meant that only the electrons within a distance d=vt=v (since t=1 second) from the surface, at the beginning of that particular second, will make it through the surface before the second ends. In other words, the electrons in a volume vdA will cross the surface of area dA during the second. If n is the number of mobile electrons in every unit volume of the conductor, the total number of electrons in a volume vdA will be nvdA.

Electric current

Electrons crossing the area dA

With the number of electrons crossing known, the total charge that crosses the elemental surface dA in a second, is simply found to be (-e)nvdA (-e, as usual, is the charge of each electron). Hence, the current passing through dA is dI=-envdA. Now, recall that we’d defined the electric current density as the amount of current flowing per unit area J=\frac{dI}{dA}.
With that, we may write, the electric current density through the conductor to be,

    \begin{align*} J=&\frac{dI}{dA}=\frac{-envdA}{dA}=-env\\ \mathbf{J}=&-en\mathbf{v} \end{align*}

But we’ve already seen in equation (2) that \mathbf{v}=-\mu \mathbf{E}.
So, 

    \[\mathbf{J}=&en\mu \mathbf{E}\]

 Again, we’d defined the mobility as, \mu=\frac{e\tau}{m*}. Hence we get,

    \[\mathbf{J}={e^2n\tau \over m*} \mathbf{E}\]

Calling {e^2n\tau \over m*} the electrical conductivity \sigma, we have arrived at equation (1),

    \[\mathbf{J}=\sigma\mathbf{E}\]

 If the mean free time \tau and the effective mass m, do not depend on the applied electric field (which is the case when the current is not too high, causing the temperature to rise), we can safely say that the electric current density is directly proportional to the applied electric field. This statement, connecting \mathbf{J} and \mathbf{E} is a form of the famous Ohm’s law, which we will explore further in the next article.

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Roshan Sawhil

Roshan Sawhil is a Physics postgraduate who rejoices both doing and explaining Physics. He also finds doing Philosophy as a leisure activity quite interesting. You can find and connect with him on Facebook and Twitter.

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