- Example: Infinite, straight current carrying wire
- Example: Magnetic field of a coaxial cable
- Example: Magnetic field of a perfect solenoid
- Example: Magnetic field of a toroid
- Example: Magnetic field profile of a cylindrical wire
- Example: Variable current density
7.3 Ampere’s Law
Since there are many similarities between electric field and magnetic fields, we adopted an approach to the magnetic field cases, as we did earlier for the electric field, which we have seen that the source of the electric field was a charge which generated its own electric field. In turn, if we place any other charge inside of this electric field, that electric field, as an external electric field, exerted a force on that other charge.
The first part of these interactions, in other words, how the electric field was generated from its source, was first calculated through an experimental law, which we called it Coulomb’s Law. It was an experimental law, and it was equal to, in its most general form, as a constant, Coulomb constant, 1 over 4πϵ0 times integral of dq over r2 times the unit vector r̂ pointing along the direction of this position vector r. And, of course, this was for the charge distribution’s case which was treated such that it was divided into incremental, very small charges, and each one of those charges were treated like a point charge.
Later on we have studied the point charge formations, specifically an electric dipole. If we recall that, that was a system which was consisting of two equal magnitude but opposite sign charges separated by a very small distance from one another.
We have calculated the electric field of such a system along the axis of the dipole, some z distance away from the dipole. For the case of z much, much greater than d, we have found that electric field, which was equal to a quantity we defined as magnetic dipole moment vector divided by 2πϵ0z3. The magnetic dipole moment vector was such that, or the magnitude of this vector was such that the magnitude of the charge of the dipole times the separation distance, and it pointed always from negative to positive charge.
Alright. We have seen that the source of magnetic fields is a moving charge, qv, or moving charges, electric current, which generates the associated magnetic field, and again, we will see a little bit later if we place any other moving charge or a current carrying wire inside of this magnetic field, let’s see, q other v, we will see that that external magnetic field relative to this moving charge will generate a force on that moving charge or current carrying conductors.
We started to analyze the first part of this interaction, how the magnetic field is generated from its source, and that was calculated, like in the case of electric fields, by means of an experimental law which was called Biot-Savart Law. The mathematical form of this law for a single charge was given as B is equal to μ0 over 4π qv cross unit vector along position vector r divided by r2.
Here, if we have a charge moving at a certain velocity and r represented the position vector drawn from charge to the point of interest and unit vector r̂ is the unit vector along that direction. And that was the mathematical form of the Biot-Savart Law, so in a way that corresponds to the Coulomb’s Law in the case of electrical interactions.
Then we went ahead and calculated the magnetic field of a current loop, specifically a circular current loop along its axis, and we have seen that when we consider such a loop carrying a current i, the magnetic field along the axis that it generates, for this case, is in vertical direction, pointing upward direction. Again, if you represent the location of this point relative to the center as z and the radius of the loop as big R, and for z, much, much greater than big R case, we have found out that the magnetic field is equal to μ0 over 2π, times a quantity which we defined as magnetic dipole vector μ divided by z3.
And here the magnetic dipole moment vector μ was defined as N times i times A where N was the number of turns if we have more one turn of this loop. Current i represents the current flowing through these turns, and A is the surface area vector surrounded by this current loop. From these expressions you can see the similarities, again, between the electric field and magnetic field by comparing these expressions with one another.
Again, as we were studying the electrical interactions, from the concept of electric flux we have defined another very useful law, which we called it Gauss’ Law. In order to calculate the electric field for some special cases, and the law was in the form of E dot dA integrated over a closed surface is equal to net charge enclosed inside of the volume of this closed surface s divided by ϵ0, permativity of free space.
Well, we have a corresponding case for the magnetic interactions, and the corresponding law in this case is known as Ampere’s Law. The mathematical form of this law is given as integral of B dot dl, magnetic field dotted with an incremental displacement vector over a closed contour, or closed loop, is equal to μ0 times i-enclosed (ienc) where ienc is the net current flowing through the area surrounded by this closed loop c.
We’re not going to go over the derivation of Ampere’s Law, and if we look at this expression, like in the case of Gauss’ Law magnetic field is inside of this loop integral. To be able to calculate the magnetic field from this law, we should be able to take the magnetic field outside of this integral, and to be able to do that, as you know, in explicit form, B dot dl is going to be equal to B magnitude times dl magnitude times cosine of the angle between these two vectors.
So that will bring a requirement that in order to apply this law we have to find a closed loop, a hypothetical loop ,like in the case of Gauss’ Law that we were after a closed hypothetical surface such that along that loop the magnitude of the magnetic field would remain constant all the time. Furthermore, again, the angle between B and dl also should remain constant all the time.
Okay. Let’s look at this Ampere’s Law now in detail. The mathematical form is the integral of B dot dl over a closed contour, or a closed loop, is equal to what we call μ0 ienc. Here, if we consider hypothetical loop of c, dl represents incremental displacement vector along this loop. Therefore dl is the incremental displacement vector along the closed loop c that we choose.
ienc is the net current passing through the area surrounded by this loop. In other words, we’re talking about this area, and we’ll look at the net current passing through this area.
For example, if I have a bunch of currents, i1, i2, i3 coming out of plane, and i4, i5 going into the plane, the net current will be i1 plus i2 plus i3 if I choose out of plane as my positive direction minus i4 plus i5. So, ienc is going to be i1 plus i2 plus i3 minus i4 minus i5 for this case. And if I have a bunch of currents outside of this loop, let’s say i6 and i7, and these are of interest,
again, if you recall from Gauss’ Law, when we chose this hypothetical closed Gaussian surface, q enclosed represented the net charge inside of the regions surrounded by that surface. Any charge outside of that surface was of interest. So, similar to that case, now we’re dealing with a closed loop which will surround a surface in this case, and the net current passing through the surface surrounded then by that loop is what we define, or call it as ienc. Any current outside of that region will be of interest.
Another point that one has to be very careful over here is, again, learning what directions that these vectors representing. For example, the incremental displacement vector over here, dl, is representing a displacement along this hypothetical loop that we choose. On the other hand, we have displacement vector when we express the Biot-Savart Law — let me rewrite that over here — which its mathematical form was given as that the incremental magnetic field generated by an incremental current element is equal to μ0 over 4π times i dl cross r over r3.
So here we have this incremental displacement vector, dl, in Biot-Savart Law, but this represents a completely different displacement vector than the dl inside of the Ampere’s Law. Here, if a current is flowing through a wire, something like this, and let’s say, in this direction, i dl, or the dl over here, is an incremental displacement vector in the direction of flow of current along the conductor, along the wire, and r is the position vector drawn from this to the point of interest.
So here dl is in the directional flow of current, whereas in this case dl does not have anything to do with the directional flow of current, but it is along the hypothetical loop that we choose.
Okay. Now, if we express the Ampere’s Law, in explicit form, which will be equal to B magnitude, dl magnitude, times cosine of the angle between these two vectors integrated over this closed loop c, which is equal to μ0 times the net current passing through the area surrounded by this loop c.
To be able to take advantage of this loop, I mean, this law, to calculate the magnetic field, we should be able to take this magnetic field outside of this integral. In order to do that, the magnitude of the magnetic field should remain constant along this closed loop that we find. Furthermore, the angle θ between vector B and dl should also remain constant all the time.
Like, again, similar to the case of electric field, as we have seen in that case that the Coulomb’s Law was applicable to all cases, but a disadvantage was the integration, sometimes we end up with complex integrals, hard to take, and sometimes we end up by taking them only numerically through computers. Like in this case, also, Biot-Savart Law is applicable for all cases, but here, again, the Biot-Savart Law, sometimes, depending upon the current geometry might be complicated integration, and therefore, it ends up with a similar type of disadvantage that we faced when we were applying Coulomb’s Law.
On the other hand, Ampere’s Law, like Gauss’ Law, here, is going to be a lot easier to apply in comparing to the Biot-Savart Law, but its disadvantage will be, again, its limited application, because only for some special cases we’re going to be able to find a closed hypothetical loop such that the necessary conditions will be satisfied in order to apply Ampere’s Law.
So, if we summarize these, we can say that Ampere’s Law is easy to apply in comparing to Biot-Savart Law, but it has limited applications. In order to apply Ampere’s Law one has to find a closed loop such that, the first condition is magnitude of the magnetic field vector, which is B magnitude, should remain constant along the loop all the time. Let’s say this loop is also referred as “Amperian loop”.
And, second condition, the angle between the magnetic field vector B and incremental displacement vector dl, should remain constant all the time. Again, very similar to the cases of Gauss’ Law, these two conditions, the condition one and condition two will imply a certain symmetry in order to apply Ampere’s Law.