Monday, October 13, 2014

How To Solve Physics Problems Maxwell's Equations problems and solutions

Maxwell's Equations

 The discussion of Maxwell's equations starts with four basic laws of electromagnetism, each of which has been described in a previous chapter. These laws can be described in words, but a full appreciation of them requires a good calculus background. Whether you have sufficient calculus background to fully appreciate the scope of these equations or not, our discussion, which will use both words and equations, should be helpful to you.

 Four Basic Laws of Electromagnetism.

The first basic law is Gauss' law for electricity. imageThis law states that the summation of the electric vector over a closed surface times the vector representing that surface must add up to the total charge contained. Visualize a sphere, or irregular volume if you like, containing charge. On each element of this surface there is an electric field vector and a vector ds normal to the surface representing the element of surface. The dot product of these two vectors summed over the surface must equal the charge contained.

The second basic law is Gauss' law for magnetism. imageThe integral in this law is the same form as Gauss' law for electricity. Again visualize a surface but with B representing the magnetic field on the surface with the dot product of B and ds being summed over the surface. In this case, however, the summation (integral) is equal to zero!

The third basic law is Faraday's law of induction. imageThis law states that magnetic flux, this construct visualized as magnetic field lines, changing with time through a closed area equals the sum of the dot product of the electric field and the vector representing the loop (around the area). Visualize a loop of wire normal to a changing magnetic field. An emf and current is induced in the wire when the flux (of magnetic field) through the loop changes.

The fourth basic law is Ampere's law. image

The sum of the dot product of the magnetic field and a vector representing an element of a loop around the current is equal to the current contained. A simple visual picture is a wire carrying a current with a circle symmetric about the wire and in a plane normal to the current. The field is constant on the circle, so this integral reduces to the constant field times the length of the circle. This integral is proportional to the current.

These four equations are the basic laws of electromagnetism. They are unique among electric laws in that they are valid for both stationary and rapidly moving charges. Now let's take a look at these equations from a symmetry point of view remembering that εo and μo operationally serve only to define a system of units.

Notice that equations 42-1 and 42-2 are symmetric in that they are surface integrals, but the first is equal to charge and the second to zero. This difference implies that there are discrete charges but no discrete magnetic poles. Equation 42-4 describes a current of charges, but there is no corresponding current of magnetic monopoles evident in equation 42-3. This is as we would expect, since based on equation 42-2 there are no magnetic monopoles.

Looking at equations 42-3 and 42-4 another asymmetry is the lack of a time varying electric field term in equation 42-4. If there are no magnetic monopoles then we expect no term corresponding to a flow of monopoles in equation 42-3, but the question of a time varying electric field term in equation 42-4 is not so easily dismissed. Time varying electric fields do exist.

Look at the consequence of a d Φ E/dt term in equation 42-4. To be dimensionally correct d Φ E/dt) would have to be multiplied by a constant with the same units as μoεo. Note that εo(d Φ E/dt) has the units of current. Adding the term μoεo(d Φ E/dt) produces Ampere's law as modified by Maxwell. imageThis completes our discussion leading to Maxwell's equations as summarized below. image

image

The Displacement Current

The most convenient way to study the time varying electric field term is with a parallel plate capacitor. Figure 42-1 shows a parallel plate capacitor with current to the plate and a growing electric field producing a magnetic field. Note the direction of the magnetic field for the displacement current is the same as for the physical (or real) current.

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Fig. 42-1

 For the connecting wire. Ampere's law is valid,image For the space between the parallel plates the physical current is zero, so the Maxwell added term in Ampere's law is valid andimage Both integrals can be evaluated over a circular path with radius equal to the radius of the parallel plates. imageThe electric flux is simply EA, soimage and equating the physical current to the displacement current imageThis is the magnetic field at the edge of the capacitor. This is a small field and a transient one. The dE/dt cannot be increased rapidly for a long time! A magnetic field due to this “additional term” in Ampere's law was discovered over 60 years after it was first predicted by Maxwell.

The quantity imageis known as the displacement current and the Ampere-Maxwell equation is sometimes written as image

In the case of the parallel plate capacitor the displacement current is equal to the physical current to the plates.

42-1 Find the displacement current and the magnetic field in a parallel plate capacitor with circular plates of radius 5.0 cm at an instant when E is changing at 1.0×1012 V/m.s.

Solution: The displacement current is imageThe magnetic field is image

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42-2 Find the maximum magnetic field at the edge of a parallel plate capacitor with radius 6.0cm and separation 2.0cm connected to a sinusoidal voltage source of 1000 Hz and 200V maximum.

Solution: The basic equation isimage but for this case imageso The voltage is ν=Vo sinωt and andimage

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