Ch0235 - Chapter 35 Gausss Law for the Magnetic Field, and,...

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Chapter 35 Gauss’s Law for the Magnetic Field, and, Ampere’s Law Revisited 309 35 Gauss’s Law for the Magnetic Field, and, Ampere’s Law Revisited Gauss’s Law for the Magnetic Field Remember Gauss’s Law for the electric field? It’s the one that, in conceptual terms, states that the number of electric field lines poking outward through a closed surface is proportional to the amount of electric charge inside the closed surface. In equation form, we wrote it as: We called the quantity on the left the electric flux . Well, there is a Gauss’s Law for the magnetic field as well. In one sense, it is quite similar because it involves a quantity called the magnetic flux which is expressed mathematically as and represents the number of magnetic field lines poking outward through a closed surface. The big difference stems from the fact that there is no such thing as “magnetic charge.” In other words, there is no such thing as a magnetic monopole . In Gauss’s Law for the electric field we have electric charge (divided by e o ) on the right. In Gauss’s Law for the magnetic field, we have 0 on the right: (35-1) As far as calculating the magnetic field, this equation is of limited usefulness. But, in conjunction with Ampere’s Law in integral form (see below), it can come in handy for calculating the magnetic field in cases involving a lot of symmetry. Also, it can be used as a check for cases in which the magnetic field has been determined by some other means. Ampere’s Law We’ve talked about Ampere’s Law quite a bit already. It’s the one that says a current causes a magnetic field . Note that this one says nothing about anything changing. It’s just a cause and effect relation. The integral form of Ampere’s Law is both broad and specific. It reads: (35-2) where: the circle on the integral sign, and, l h d , the differential length, together, tell you that the integral (the infinite sum) is around an imaginary closed loop . B h is the magnetic field, l h d is an infinitesimal path element of the closed loop, μ o is a universal constant called the magnetic permeability of free space, and I THROUGH is the current passing through the region enclosed by the loop. o ENCLOSED Q = A d E h h A d E h h = Φ E THROUGH o I = l h h d B 0 = A d B h h A d B h h = Φ B
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Chapter 35 Gauss’s Law for the Magnetic Field, and, Ampere’s Law Revisited 310 What Ampere’s Law in integral form says is that, if you sum up the magnetic-field-along-a-path- segment times the length of the path segment for all the path segments making up an imaginary closed loop, you get the current through the region enclosed by the loop, times a universal constant. The integral on whatever closed path upon which it is carried out, is called the circulation of the magnetic field on that closed path. So, another way of stating the integral form of Ampere’s Law is to say that the circulation of the magnetic field on any closed path is directly proportional to the current through the region enclosed by the path. Here’s the picture:
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Ch0235 - Chapter 35 Gausss Law for the Magnetic Field, and,...

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