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Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
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16.18 | Maxwell Equations |
In this Physics tutorial, you will learn:
Topics related to electricity and magnetism are numerous; they include a wide range of concepts and equations, sometimes quite difficult to memorize. For this reason, the Scottish scientist James Clerk Maxwell summarized them only in four equations, known as Maxwell Equations, which represent the key relationships between quantities in Electromagnetism. In a certain sense, Maxwell Equations represent a wonder of scientific synthesis ability.
As explained in tutorial 16.1 "Introduction to Magnetism", all magnets have two poles: one north and one south. The magnetic field lines are closes; they start from the north-pole and end at the south-pole of the same magnet. As and evidence for this, we can consider the shape the small iron filings poured on a paper sheet take when a magnet is placed below the sheet.
We have also provided (in the same tutorial) the definition of magnetic dipole, which is the smallest possible magnet made from a proton-electron pair. This means the magnetic monopoles do not exist.
The Gauss Law for the magnetic field implies that magnetic monopoles do not exist. Mathematically, this law means that the net magnetic flux Φm through any closed Gaussian surface is zero. The Gauss Law formula for magnetic field is
This outcome is different from the Gauss Law in electric fields. Recall that in electric fields, we have
Unlike in electric fields in which the net electric flux through a closed Gaussian surface is proportional to the charge, in magnetic field the net magnetic flux through a Gaussian surface is zero. This is because the net magnetic charge in all magnets is zero because the charge is balanced; in magnets, we simply have a certain regular alignment of dipoles but the net charge is zero.
The figure below gives a clearer idea on this point.
As you see from the figure, there are two lines entering the loop and two lines leaving the loop. Therefore, the net flux flowing through the Amperian loop is zero.
In which of the points shown in the figure
In the tutorial 16.7, we explained that a changing magnetic flux induces an electric field in a loop, which on the other hand generates an emf (and current) in the loop.
This represents the Faraday's Law and its integral form is
The reverse is also true. A changing electric flux can induced a magnetic field. The equation representing this property is known as Maxwell Law of Induction and the corresponding equation is
Here B⃗ represents the magnetic field induced along a closed loop by means of the changing electric flux ΦE in the region enclosed by the loop.
For example, let's consider a parallel-plate capacitor with circular plates, the lateral and front view of which, are shown below.
The front view shows the right plate of the capacitor viewed from inside the plates.
If we assume the charge in the capacitor places increases at a steady rate through a constant current I in the conducting wire, a changing electric flux will occur in the plates - a process that will induce a magnetic field on the capacitor plates because the electric field between the plates changes at a steady rate as well. This induced magnetic field is circular because we may assume any Amperian loop as the border of a current-carrying wire and when using the right hand-rule, we obtain a circular magnetic field if the current carrying wire is straight.
In other words, if the circles having the radii r and R are through as radii of two current-carrying wires of different thickness, the corresponding magnetic fields will be circular, and their directions are found using the right-hand rule. In the specific case, since the current is inwards, the direction of thumb will be onto the page, while the other curled fingers will be oriented clockwise, as shown in front view of the above figure.
The same thing can be said when a circular (Amperian) current-carrying loop is taken inside a uniform magnetic field B. In this case, we obtain the following figure:
The only difference to the previous figure is that the direction of electric field here is anticlockwise.
The last two equations shown earlier are similar, as they both involve a closed integral (the integral taken along a closed loop), the result of which, involves the change of the other flux in the unit of time (B → dΦE and E → dΦB).
In other words, an increasing electric field E directed onto the page, induces a counter-clockwise magnetic field B in the loop, while an increasing magnetic field B directed onto the page, induces a counter-clockwise electric field E in the loop.
Combining the last two equations, we obtain the Ampere-Maxwell Law
As special cases, we can consider the two following scenarios:
When there is a current but no change in electric flux (such as a wire carrying a constant current), the first term on the right side of the Ampere-Maxwell Law is zero. As a result, the Ampere's Maxwell Law reduces to Ampere's law of Induction we have seen in tutorial 16.6.
When there is a change in electric flux but no current (such as inside or outside the gap of a charging capacitor), the second term on the right side of the Ampere-Maxwell Law becomes zero, and so the Ampere's Maxwell Law reduces to the Maxwell's Law of induction
Thus, from the expression derived in (a), we obtain for the magnetic field B at the given distance r from the centre of capacitor plates:
When analysing the equation derived from the Ampere-Maxwell Law
it is obvious that the expression
must have the dimension of current. This current is known as the "displacement current, id", despite no current is really being displaced. Thus, the above equation becomes
Let's consider again the charging capacitor of the previous section, shown in the figure below.
Let's find a way to relate the real current I that is used to charge the capacitor plates to the fictitious displacement current Id which is associated to the change in the electric field between the plates.
From the Gauss Law for electric field
we obtain for the charge Q on the plates at any time:
where E is the magnitude of electric field between the plates at that time.
Differentiating the above equation with respect to time, we obtain for the real current I which charges the capacitor:
As for the displacement current Id, we have
As you see, we obtained the same expression for the displacement current to the expression obtained for the real current. Thus, we can consider the fictitious displacement current Id simply as a continuation of the real current I from one plate of capacitor to the other plate flowing through the gap between plates. Although no charge actually moves across the gap between the plates, the idea of the fictitious current Id can help us to quickly find the direction and magnitude of an induced magnetic field as we will see in the next paragraph.
In tutorial 16.2 "Magnetic Field Produced by Electric Currents" we explained that the direction of magnetic field produced by a current-carrying wire is found by using the right-hand rule, i.e. we grasp the wire with the right hand outstretching the thumb in the direction of current. In this case, the other four curled fingers show the direction of magnetic field.
We can apply the same rule for the magnetic field produced by the displacement current as well. In this case, we consider the cylinder formed by the capacitor plates and the space between them as a cylindrical conducting wire of radius R. Hence, applying the known equations derived in the tutorial 16.2, we obtain for the magnetic field produced by the displacement current inside the capacitor plates, at a distance r from the central axis (r < R),
and for the magnetic field outside the plates (at r > R from the central axis),
A circular parallel-plate capacitor with plate radius R = 4 cm is being charged by a current I = 5A.
What is the magnitude of ∮B ∙ dL at a distance of R/7 from the central point of capacitor plates?
What is the magnitude of magnetic field at this distance?
First, let's calculate the magnitude of the displacement current in the given section between the capacitor plates. Since the magnitude of the displacement current Id represents a part of the real current I which charges the capacitor, the ratio between the area encircled by the loop of radius r to the total area between the plates is equal to the ratio of the two abovementioned currents. Thus, we have
Hence, the displacement current at the given position is
Therefore, since at the given distance
we obtain for the value of integral
Since r = R/5 represents a location inside the plates, we have for the magnetic field B:
The four equations shown in the table below are known as Maxwell Equations. As explained at the beginning of this tutorial, these equation represent the key relationships between quantities in Electromagnetism. Thus, the four Maxwell Equations expressed in the integral form, are:
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