 # Electric dipole

Learning objectives

1. Outcome one
2. Outcome two

Display equations ?

# Electric dipole

By Roshan Sawhil

Electrically charged objects push and pull each other for the very reason that they possess electric charge. If we consider two charged objects and observe that they pull (attract) one another, we say the objects have opposite kinds of charge. If one is positive, the other is negative. When we release them, they’ll accelerate towards each other. On the other hand, if we observe a repulsion between the objects, we say they possess the same kind of charge – either both are positive or both are negative. Such charges will fly away from one another, if set free.

But what if we bring together two opposite charges separated by a fixed small distance, perhaps by putting them at two ends of a rigid rod, so that they aren’t allowed to move any closer due to the attraction. This is essentially the idea behind the electric dipole. Before we explore the dipole further, let us learn a few concepts that’ll help us in our understanding.

Imagine any empty space stretching infinitely in all directions. We now place a point positive charge $Q$, which we shall refer to as the source charge hereafter. Due to the presence of this charge, the empty space is no longer entirely empty. The space around the source charge is filled with its influence. How do we test this influence? Bring in another point positive charge $q$, which we shall call the test charge. Placing it at different points in the vicinity of the source charge, we will observe that the test charge experiences a repulsion pointing directly away from the source charge. And this repulsive influence pervades all space – wherever you place the test charge, it will experience a repulsion (large or small, depending on how far it is from the source).

Now since the force between the source and the test charges is repulsive, it is easier to move the test charge away from the source and harder to move it towards. By this we mean that an additional push must be applied on the test charge in order to move it towards the source (against the repulsion). Let us, starting from a point A, move the test charge closer to the source charge, to a point B. Now, we can cause this displacement through an unlimited number of paths (connecting A to B). And we might intutively expect that the total force we must apply on the test charge to move it from A to B will be different for different paths – the smallest for the shortest path and largest for the longest.

It however turns out that no matter what path we take, the total force that must be applied to the test charge to displace it from point A to point B is the same. Whether you move the test charge from A to B along a straight line connecting the two points or along a very long path winding around the universe several times, the total force you must apply is the same. We therefore say the work done here is path independent (recall that work done is the product of the part of the force acting in the direction of the displacement and the displacement). To understand this point better, think of this: every time you move the test charge directly towards the source charge, you are doing work. But every time you move it directly away, the repulsive force from the source charge is doing the work. Any random path can be divided into a series of movements directly towards and away from the source. Any movement in the lateral direction (perpendicular to the line joining the test and the source charge) will involve no work.

If you are to measure the total work done by you alone, you will consider only those parts of the movement that were directed straight towards the source. And this is unique independent of what path you took. For example, as you can see in the figure, in moving the test charge from point A to B, you could divide the entire path into a radial (straight towards the source) and an angular (along the arc of the circle centred at $Q$) part. You must realise that once the points A and B are fixed, the length of the radial part is the same no matter what the actual path taken is, by the test charge. And the raidal part alone contributes to the over all work done.

This path independence of work allows us to associate a unique number for every pair of the points A and B, so that that number doesn’t depend on the path connecting the two points (along which the test charge was moved) but only on the locations of A and B (the initial and final positions of the test charge). We call this number the electric potential difference. If the initial point A is located infinitely far from the source, the repulsive force on the test charge here is essentially zero. Then the work done on the test charge in moving it all the way from A to B will also be path independent and will depend only on the location of point B. We then speak in terms of the electric potential at point B. It is a number associated especially with the point B and depends only on how strong is the source charge $Q$ and how far from it is the point of interest B. For a fixed source charge, the potential takes on a small value if the distance ($r$) from the source is large and a large value as we move closer to the source. Mathematically, electric potential is,

$V=-k{Q\over r^2}$

Now, without digressing too much, let us consider the electric dipole and apply the concept of electric potential for the dipole. Firstly the dipole is by definition a combination of two equal and opposite charges $+q$ and $-q$, separated by a small distance $d$. We wish to test the influence of this dipole on the space around it, by measuring the electric potential at a random point P (say) in the vicinity of the dipole. For this, we bring in a positive test charge $q$, place it at P and measure the work done on it in bringing it from infinity (a place very far off from the dipole).

Clearly, the positive charge $+q$ of the dipole repels the positive test charge while the negative charge $-q$ attracts it. So you might think the attraction and the repulsion cancel one another, and electric potential at any point around the dipole is zero. But, as we noted earlier, the strength of the attraction and the repulsion depends on how far the measurement is being made from the source charge. Since there is a finite distance $d$ between the charges of the dipole, the distance of these charges from the test charge needn’t be equal in general and hence while at some points that are closer to the positive charge than they are to the negative charge, the repulsion dominates while at those closer to the negative charge, the attraction dominates. There indeed are special points that are equidistant from both the charges of the dipole, where the electric potential is zero. They are nevertheless special points.

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