2.5 Dipole in an External Electric Field
Let’s now consider an electric dipole in an external electric field. All right, so far we have looked at the interactions such that a source charge generates its own electric field. And we calculated this electric field through Coulomb’s Law. And it was such that, in its most general form for a distribution, integral of dq over 4 Pi Epsilon zero r squared. And we also mentioned earlier that if you place any other charge, q other, into a region of this electric field, that electric field will exert Coulomb force on that other charge. In other words, that force, Coulomb force, was equal to this other charge times the electric field generated by the source.
The electric field here becomes an electric, external electric field, for this other charge. Of course, these are one to one type of interactions. This other charge will generate its own electric field and that electric field will become an external electric field for the source in this picture. Then that too will exert a force, Coulomb force, on that charge.
Again, so far we have been dealing with this part of the interactions. In other words, how the electric field is generated from its source.
Now, we’re going to look at an example that a charge, placed charge or charges, placed inside of an external electric field. And for that we will use an electric dipole. As you recall, an electric dipole is a system with two point charges, equal magnitudes and opposite signs separated by a very small distance. And we will place a dipole in an external electric field.
Let’s assume that we have an external electric field and a region of interest pointing to the right, in this form. And it’s a uniform electric field. Uniform, external electric field. Replace a dipole inside of this region, an electric dipole, with positive q charge separated from the negative q by a small distance of d, separation distance.
As soon as we place these charges into this external electric field, that electric field will immediately exert Coulomb force on these charges. For the positive charge, that force is going to be equal to q times the electric field. Therefore, it’s going to be pointing in the same direction with the electric field, external electric field. And that’s going to be equal to q times the electric field vector.
Similarly, negative charge will also be under the influence of Coulomb force generated by this external electric field. And that is going to be equal to minus qe. Therefore, it will be pointing in opposite direction to the direction of the electric field. Here, the force on this charge will be minus qe.
Since these two charges have the same magnitude, the magnitude of these forces will be the same. But if we look at the orientation of these forces, we can easily see that under the influence of these two forces, if we pull the system with a force f to the right and pull it, minus q, with a force of f to the left, then we will cause a rotation.
In other words, the system is going to rotate in clockwise direction. And as you recall, the physical quantity which causes a rotation is called torque. It means that this rotation will imply an existence of a torque.
We’re going to now try to determine the magnitude and direction of this torque. Naturally this system will rotate about an axis which is passing through its center. And as you remember, the torque was defined as position vector cross with force vector.
Position vector r, is defined as, it is drawn from the axis of rotation to the application point of force. We need to know the direction of these two vectors that we’re taking cross product, to be able to determine the direction of the resultant vector. As you remember, when we take the cross product of two vectors, we end up with a new vector. In other words, a quantity, a physical quantity with both magnitude and directional properties.
And to be able to determine the direction of the resultant vector, if the original vectors are not given in terms of their components with respect to a coordinate system, the only way that we can determine the direction of the resultant vector from a cross product is by applying right hand rule. And the right hand rule says that we hold our right hand fingers in the direction of first vector, in this case the position vector r, and then curl them towards the second vector while keeping the thumb, right hand thumb, in up position. And during that process the direction of thumb will give us the direction of the resultant vector.
So in order to apply right hand rule we need to know which direction that r is pointing and which direction that f is pointing. Of course, from our diagram, we know which direction that f is pointing. And the r from this definition points from axis of rotation to the application point of the force.
Let’s denote this force with f sub plus and this one with f minus, associated with the positive and negative charges respectively. And the position vector of this force, therefore, will be pointing from the axis of rotation to the application point of the force. And for the f minus, it is going to be pointing from the axis of rotation to the application point, therefore in this direction.
Now applying right hand rule, we will first hold our right hand fingers in the direction of this vector, which is pointing basically this way, and then curl them towards the second vector and that is f plus pointing to the right. So, if you hold your right hand fingers in this direction first and then adjust them so that you can curl towards the second vector f plus and in doing that you will see that keeping the right hand thumb up, that thumb will be pointing into the plane.
Therefore, the torque generated by f plus relative to this axis of rotation is going to be, less than or that by Tau plus, pointing into the plane. And we will denote that direction by cross sign like this. You can visualize this as if an arrow going into a plane and we see its tail, here on the surface of the plane.
Similarly, if we look at the other force, now its position vector is pointing this way. And if you hold your right hand fingers in this direction and adjust them so that you can curl toward the second vector, that is f minus pointing this way, so if you try to curl them this way then you will see that your right hand thumb again for this case also pointing into the plane of the screen.
So Tau minus will also be pointing into the plane. Indicating that these two torques, generated by these two forces will cause a rotation in the same direction because they are both in the same direction. Net torque will be, therefore, the vector sum of these two, since they are in the same direction, we will just add their magnitudes.
So, torque total is equal to torque due to the positive charge, I mean due to the force acting on the positive charge, plus torque due to the force acting on the negative charge. Tau plus plus Tau minus. And since the magnitude of the forces are equal and as well as the magnitude of the position vectors r plus and r minus are equal, indicating that the magnitude of these two torques, Tau plus and Tau minus.
Therefore, from here, we can say that Tau plus, magnitude, is equal to Tau minus, magnitude. So, the magnitude of the torque is going to be equal to twice of either Tau plus or Tau minus, since they have equal magnitudes.
Okay, if we look at our diagram, we see that the dipole is making a certain angle with the electric field vector and let’s denote that angle by Theta. That angle is Theta, then this angle will also be Theta, and as well as this angle will also be Theta.
Well, as you recall from the cross product properties, the magnitude of Tau is equal to the magnitude of r times magnitude of f times sine of the angle between these two vectors. So if we look at the Tau plus, which is equal to r plus crossed with f plus, then we are talking about the magnitude of r plus times magnitude of f plus times the sine of the angle between them, and this is another important property that we have to remember.
When we talk about the angle between two vectors, we’re talking about the angle that they make whenever their tail coincides with one another. In other words, the angle between r plus vector and f plus vector is not this angle. We simply carry the r plus vector such that the tails of these two vectors coincides and then measure the angle between them. That is the angle which is defined as the angle between these two vectors. And in our diagram that angle is nothing but angle Theta.
So, Tau plus, magnitude, will be r plus magnitude times f plus magnitude times sine of the angle between these two vectors. Well, f plus magnitude, r plus magnitude, after the magnitude of the position vector, is this distance. And that distance is half of the dipole separation distance.
So here, r plus will be equal to d over 2. And f plus is equal to the positive charge times the electric field vector magnitude. Therefore, Tau plus becomes equal to d over 2 times e times q. So, d times q times e times sine of the angle between these two vectors.
Then, total torque, since it is magnitude times twice of the torque generated either by f plus, force on the positive charge, or force on the negative charge, multiplying this quantity by 2, we’re going to end up with d times q times e sine Theta.
Here, when we look at this expression which is the total torque or net torque on that dipole, causes the rotation of the dipole, we see the product of the separation distance and the magnitude of the charge on the dipole. Now these two quantities are unique properties of a specific dipole. In other words, every dipole will have their own unique charge and their own unique separation distance.
So, this product will also be a unique quantity for a given dipole. And we have a special name for that product, it is called electric dipole moment. Actually, this is a vectoral quantity. And we will denote this by p and its magnitude is equal to qd. Direction of dipole, moment vector, is such that it points from negative charge to positive charge, in a given electric dipole.
Therefore, relative to this directional definition, electric dipole moment vector p is going to be pointing from negative charge to positive charge. Something like this. And by looking at this diagram, we see that that too makes an angle of Theta with the electric field vector. And it points from negative to positive charge.
Now going back to the torque equation, since torque is equal to, now, magnitude of dipole moment vector times the electric field vector times sine of the angle between them. And this is nothing but an equation for the magnitude of a cross product.
Since Theta is the angle between electric dipole moment vector and the electric field vector, then the torque can be expressed, the net torque can be expressed as dipole moment vector crossed with external electric field vector. And that is the torque on an electric dipole.
We will look at some specific molecules. We see that some of them show dipole characteristics. In other words, as the atoms come together to make that specific dipole, they represent a positively charged ion and a negatively charged ion separated by a very small distance from one another.
So, they will all have an associated dipole moment vector. And if you take these molecules and place it in an external electric field then those dipole molecules are going to rotate under the influence of this torque and they will align along the electric field lines. And you will see or study these type of molecules if you take a chemistry course later on.