- Example: Magnetic field of a current loop
- Example: Magnetic field of an infinitine, straight current carrying wire
- Example: Semicircular wires
7.2 Magnetic Field- Biot-Savart Law
We have seen that there is a major similar behavior when we compare the electric charges or electricity with magnetism. We have seen that the like charges repel and the unlike charges attract one another from the electricity or the electrical interactions. We have also seen that like poles repel and unlike poles attract one another.
As I mentioned earlier, it is because of this similarity between these two different types of interactions. Physicists looked for a link for a long time to see whether there is a relationship between the electricity and magnetism and also initially adopted an approach which is similar to the case of electrical interactions.
As you recall, while we were studying the electrical interactions, we looked at the interactions from the point of view that the source of electric field is the electric charge and that charge generates its own electric field, let’s say, Esource. And we have seen that if we place any other charge inside of this region then this electric field generate by the source exerts a force on that other charge. Depending upon the nature of this charge we end up with either an attractive force or a repulsive force between the source charge and the other charge.
First physicists, again, by considering these attractive and repulsive nature of the forces originating from the electrical and magnetic interactions, they suggested that the source of the magnetic field is the magnetic pole, so they simply stated that magnetic pole is the source of the magnetic field and let’s say the source pole generates the associated magnetic field. They said that, “Okay, if you place any other magnetic pole inside of this region then, like in the case of electrical interactions, this magnetic fields exerts a force on that other pole and depending on the nature of the pole, this force can either be attractive or repulsive.
This type of approach immediately failed from the fact that we cannot have an isolated magnetic pole. In other words, we cannot have a north pole by itself or a south pole by itself. Therefore the source of magnetic field cannot be magnetic pole. So this approach will say that failed right away due to the fact that since there cannot be any magnetic monopole.
Of course still physicists looked for a link between electricity and magnetism furthermore due to this similarity of the attractive and the repulsive nature of the forces between the magnetic poles. As I mentioned earlier this link came up in 1819, by the Danish scientist, Hans Christian Oersted as an accidental discovery, at 1819, by the Danish scientist, Hans Christian Oersted, while he was setting up a demonstration for his students.
Oersted was setting up a simple electric circuit such that it had, of course, a power supply, let’s say a battery with positive and negative terminals and a bunch of wires connected to different components when the circuit an electric switch and also, in the meantime, a compass was just nearby to the circuit and next to the one segment of the wire of the circuit and it was naturally aligned relative to the earth’s magnetic field passing through that point. Well, once Oersted closed the switch and let the current flow through the circuit, he noticed that the compass needle changed it’s orientation. In other words, it rotated and took a new position, let’s say, something like this.
Then he turned the switch off and realized that the compass needle went back to its original orientation, and this was the first link between electricity and magnetism, because when we look at this case, we can easily realize that what’s happening over here is that once we turn the switch on and the electric charge is starting to move, the current is going to be flowing from positive terminal towards the negative terminal. So as the charges move then there has to be a new magnetic field generated in the medium, such that this magnetic field is stronger than the earth’s magnetic field, and as a result of this, the compass needle reorientates itself as tangent to this new magnetic field. And once we turn the switch off, then the current drops to 0. In other words, charges do not move in a specific direction anymore. Therefore, the magnetic field associated with that goes down to 0 and the compass needle reorientates itself, relative to the Earth’s magnetic field passing through that point.
So let’s say that this represents the earth magnetic field passing through that location and Oersted then concluded that the source of the magnetic field is moving charge or moving charges. He simply, therefore, looked at the relationship similar to the case of electric field in the form of that moving charge. We can denote this as qv a charge with a certain velocity of v, or moving charges, which is basically electric current flowing through an incremental distance displacement of dl is the source of the magnetic field. So we can introduce the subscript of “source” over here, and once this magnetic field is generate by source, then if we place other moving charge or charges, current carrying conductor, into this region, then this magnetic field will exert a force on that other moving charge or moving charges, electric current, along that specific conductor or wire.
We will analyze both sides of these interactions and first we are going to look at how the magnetic field is generated from its source. In other words, this part of the relationship. If you recall from the electrical interactions we have seen that the source of the electric field was the electric charge. We calculated how the electric field is generated from its source through an experimental law, which was called Coulomb’s Law.
As in that case we have an experimental law for the magnetic interactions in order to calculate the magnetic field, how it is generated from its source and it’s known as Biot-Savart law. For a single moving charge, let’s say, for moving point charge, the mathematical form of this law, this experimental law, is such that B is equal to a constant μ0 over 4π times qv cross unit vector in radial direction r̂, r̂ divided by r2.
So in this case we have, let’s say, a point charge. Let’s consider a positive point charge, which is moving at a specific velocity of v, and if our point of interest is located somewhere over here, P is the point of interest where we would like to determine the magnetic field that this moving charge generates. The r is the position vector, drawn from charge to the point of interest and r̂ unit is the unit vector in that direction.
Therefore the mathematical form of the Biot-Savart Law for a moving point charge is in this form, and again, let’s make a note of this. This law is an experimental law. It is obtained as a result of a series of experiments.
As we can easily see, it involves a vector cross product so the direction of the magnetic field vector, which is generated by this moving charge and that is moving with a velocity of v, is going to be equal to the direction of qv cross r and again, as you recall, we said that if the vectors are not given in their component form with respect to a coordinate system, when we take the cross product, the only way that we can determine the direction of the resultant vector is by applying the right hand rule.
So applying right hand rule to qv cross r ,we are able to determine the direction of the resultant vector. And of course the magnitude of this vector B, is going to equal to μ0 over π times the magnitude of this cross product and that is qv magnitude times r magnitude, since it is an unit vector, it is going to be 1 in this case, and times the sine of the angle between these two vectors. That angle is the angle measured when the tails of these vectors are coinciding with another, so there is this argument over here. The angle of θ is the angle between our unit and velocity vector, so we will have sine of that angle divided by r2. That expression will give us the magnitude of this magnetic field vector.
Now, if we have more than one point charge, in other words, moving charges, which is the case corresponds to a current flowing through a wire, through a conductor, in that case, if this represents our wire for example, and the current is flowing through this wire, let’s say in this direction, then we simply generalize this expression for this case and we choose an incremental current element with a segment of this wire, of the length dl, therefore the incremental current element will be i dl.
So in other words, divide this wire into very small segments that the amount of charge moving through these segments can be treated like a incremental charge of Bq therefore the electric field associated with that segment will be equal to dB incremental magnetic field and that can be expressed as, therefore the equation can be expressed as μ0 over 4π. For q, we will have dq times its velocity times unit vector r̂, divided by r2.
Now, since current is equal to dq over dt we can express dq as i dt, and for a charge of dq to take the distance of the displacement of dl we can express its velocity as dl over dt. If we substitute these quantities in Biot-Savart law for a single point charge manipulated for an incremental charge expression we will have μ0 over 4π. And for dq we write down i dt, and for the velocity of the charge we have dl over dt cross with unit vector r̂ divided by r2.
Again, here if our point of interest is located somewhere over here, P, r̂ represents a vector drawn from the current element, i dl, to the point of interest and this is our vector, r, and unit vector r̂ ̂, is a unit vector along that direction. So here, the times dt‘s will cancel and that will leave us an expression of dB is equal to μ0 over 4π times i dl cross r̂ over r3.
Now, this is for a current carrying conductor segment of an incremental length and once we calculate the magnetic field associated with that incremental current element along the wire, then one can go ahead and calculate the magnetic field associated with next segment and then the next segment and so on and so forth. Finally, when all these segments are added to one another or all the magnetic fields, incremental magnetic fields generated by those incremental current elements added to one another then we will get the total magnetic field generated by this current carrying conductor.
Of course this addition process over here is integration and by taking the integral of both sides, in other words, we end up with a total magnetic field B is equal to integral of μ0 over 4π, i dl cross r̂ unit over r2 and here we have r2 rather than r3.
Okay, so then if we summarize the Biot-Savart Law for this two different cases and mainly for a moving point charge, this experimental law is in the form of B is equal to μ0 over 4π, qv cross r̂ unit over r2. Again, this is the case that we have a positive point charge, as an example, moving at a velocity of v, and if we are interested with the magnetic field that it generates at this point P, r is the vector drawn from this charge to that point of interest. So this is r, and r̂ is the unit vector along that direction.
Now, for the current carrying wire, in other words we consider a bunch of moving charges, electric current, then the magnetic field is equal to μ0 over 4π is constant in this expression — and I will give the information in a moment what μ0 is — we can take an outside of the integral. therefore the general form of the Biot-Savart Law for that, becomes Mu zero over four pi times integral of i dl cross r̂ unit over r2.
Here we can make one more manipulation for this case. Since a unit vector is defined as the, in a specific direction, is defined as the, for a long specific vector, is defined as the vector divided by its magnitude, because as you recall the definition of unit vector is a vector with a magnitude of one pointing in a specific direction. So, for x unit we have the unit vector î which has the magnitude of 1 and it points in the positive x direction.
For the unit vector in our direction, it will be the vector r divided by its magnitude. So we can express this as the r vector, position vector, divided by the magnitude of that vector. If you substitute this into this expression, then the magnetic field becomes equal to μ0 over 4π, integral of i dl cross r, the position vector r, divided by r2 times another r is going to come up from the unit vector r, so the whole expression is going to be equal to that μ0 over 4π, integral of i dl cross position r, divided by r3.
So once we express the magnetic field in terms of the position vector, then the denominator become r3 but this shouldn’t give us the wrong idea of that the magnetic field is varying with one over r3. Like in the case of electric field, we can see that the magnetic field is varying with 1 over r2. In other words, it’s based to the inverse square law where it’s inversely proportional to the square of the distance between the source and the point of interest. So once we express that in terms of the unit vector r̂, we can easily see that, but if we express the unit vector in terms of the position vector, then the denominator becomes r3.
Alright and in this expression, as I mentioned earlier, the terms represent the quantities such that i dl is the incremental current element, which is source that will be chosen in the direction of flow of current through the wire. r represents the position vector drawn from i dl to the point of interest, so let’s express these quantities over here. i dl is incremental current element chosen in the direction of flow of current, and r is the position vector drawn from i dl, from this current element to the point of interest.
It is extremely important that we learn what vectors these quantities are representing, i dl and r because the only way that we are going to be able to determine the direction of magnetic field is by applying, right hand rule to this cross product, i dl cross r. In order to apply the right hand rule, as you recall, we need to know the direction of each one of these vectors that we are taking the cross product. Without knowing those directions we cannot apply right hand rule.
So in this case, i dl is a vector which is pointing an incremental current element vector pointing in the direction of flow of current. As you recall, current itself is a scalar quantity. We have a flow direction associated with the current. So once we take that product of that scalar quantity with that incremental displacement vector dl and then this quantity becomes a vector and it is pointing in the direction of flow of current, and r is the position vector drawn from this element to the point of interest and these “from” and “to” words represent the direction associated with the position vector.
Since we know the directions of these two vectors that we are taking the cross product, therefore we are going to be able to apply right hand rule in order to determine the direction of the magnetic field. Since the Biot-Savart Law for a current carrying conductor is μ0 over 4π i dl cross r over r3, the magnitude of this vector B, so B magnitude is going to equal to the magnitude of the cross product over here, times these quantities, these are μ0 over 4π times the integral, magnitude of a cross product is the magnitude of the first vector, i dl, times the magnitude of the second vector which is r times sine of the angle between these two vectors. And that angle is, the angle measured when the tails of those two vectors are coinciding, which is this angle over here, so sine of that angle divided by r3.
Now we see that this r and r3 will cancel. Therefore the magnitude of the magnetic field vector will be equal to μ0 over 4π times integral of i dl, sine of θ, divided by r2. Once we integrate this over all the length of the wire, or the conductor, then we are going to be able to determine the magnitude of the magnetic field at a specific point in which its position vector is given by this position vector r.
Here now μ0 is a constant associated with the magnetic properties of the medium, and the medium in this case is either vacuum or air. It has a special name and it is called “permeability of free space”. In the SI unit system, μ0 is equal to 4π times 10 to the -7 tesla meter per amp.
And here, let’s introduce the unit of magnetic field. So the unit of magnetic field in the SI unit system is called “tesla”, and we are going to denote this by capital T. So the magnetic properties associated with the medium of interest is given by this constant, which is called permeability and permeability of free space and that is either air or vacuum, is equal to 4π times 10 to the -7 tesla meter per amperes.
So Biot-Savart Law, which is an experimental law, for a currently carrying conductor, its mathematical form is in this form and it is an experimental law. If you consider a single moving charge then the mathematical form of Biot-Savart Law is given in this form.