PES 1120 Spring 2014, Spendier Lecture 38/Page 1
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1 PES 1120 Spring 2014, Spendier Lecture 38/Page 1 Today: Start last chapter 32 - Maxwell s Equations James Clerk Maxwell ( ) Scottish mathematical physicist. He united all observations, experiments and equations of electricity, magnetism, and optics into a consistent theory. Since light is an "electromagnetic" wave (more on this in Physics 3), light also must satisfy Maxwell's equations. E&M Equations So Far Gauss s Law for E-Field : Gauss s law for electrostatics states that the electric flux through a closed is proportional to the charge enclosed. The electric field lines originate from the positive charge (source) and terminate at the negative charge (sink). qenc E E da (ε 0 = permittivity of free space) 0 Gaussian Gauss s Law for -Field One would then be tempted to write down the magnetic equivalent as Qm da 0 where Q m is the magnetic charge (monopole) enclosed by the Gaussian. However, despite intense search effort, no isolated magnetic monopole has ever been observed. Hence, Q m = 0 and Gauss s law for magnetism becomes da 0
2 PES 1120 Spring 2014, Spendier Lecture 38/Page 2 Gaussian This implies that the number of magnetic field lines entering a closed is equal to the number of field lines leaving the. That is, there is no source or sink. In addition, the lines must be continuous with no starting or end points. In fact, as shown for a bar magnet, the field lines that emanate from the north pole to the south pole outside the magnet return within the magnet and form a closed loop. Faraday s Law of Induction The most general form for Faraday s Law is: d with da (induced emf) (magnetic flux) Now, let s explore a very important and strange aspect of this. Induced EMF Let s take a very long (infinite) solenoid with a changing current. Remember that in this case, the magnetic field outside the solenoid is extremely weak. Now, outside of the conductor, let s put a conducting wire loop, with a galvonometer to measure current. It is safe to say that the magnetic field at the wire loop is zero. ut there IS a changing flux through the loop. Will there be an induced current in the loop? YES!
3 PES 1120 Spring 2014, Spendier Lecture 38/Page 3 ut what force makes the charges move around the wire loop? It can't be a magnetic force because the loop isn't even in a magnetic field. We are forced to conclude that there has to be an induced electric field in the conducting ring caused by the changing magnetic flux. This is strange since we are accustomed to thinking about electric field as being caused by electric charges, and now we are saying that a changing magnetic field somehow acts as a source of electric field. This means that even if there is no conducting ring there, there is an electric field! What is the direction of the electric force on a positive point charge? Let s look at the equation for electric potential change going from some point a to point b in the presence of a charge: b V Vb Va E ds a Now, what would we get if we turned around and returned to point a? In other words, what is the integral of E dot ds around a closed path? closed path E ds E ds V V 0 a a How would this answer change in the presence of a changing magnetic flux? d E ds This is a new relation. It states that an electric field is induced along a closed loop by a changing magnetic flux in the region encircled by that loop. This is a second way of writing Faraday's Law of induction and is one of the 4 Maxwell's equations.
4 PES 1120 Spring 2014, Spendier Lecture 38/Page 4 Correction to Ampere s Law ecause symmetry is often so powerful in physics, we should be tempted to ask whether induction can occur in the opposite sense; that is, can a changing electric flux induce a magnetic field? The answer is that it can; furthermore, the equation governing the induction of a magnetic field is almost symmetric with the above equation. d (μ 0 = permeability of free space) E ds 00 (Maxwell's law of induction) A magnetic field is induced along a closed loop by a changing electric flux in the region encircled by that loop. Example: A parallel-plate capacitor with circular plates of radius R is being charged. The change of the electric field over time is de/ = 1.50 x V/(m*s). What is the field magnitude for r = R/5 = 11.0 mm and
5 PES 1120 Spring 2014, Spendier Lecture 38/Page 5 Now recall that the left side Maxwell's law of induction, the integral of the dot product around a closed loop, appears in another equation, namely, Ampere s law: ds 0Ienc (Ampere's law) where I enc is the current encircled by the closed loop. Thus, our two equations that specify the magnetic field produced by means other than a magnetic material (that is, by a current and by a changing electric field) give the field in exactly the same form. We can combine the two equations into the single equation d E ds 0Ienc 00 (Ampere-Maxwell law) When there is a current but no change in electric flux (such as with a wire carrying a constant current), the first term on the right side is zero, and so the equation reduces to Ampere s law. 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 is zero, and so the equation reduces to Maxwell s law of induction. In 1865 (right after the American Civil War), James Clerk Maxwell was examining Ampere s Law and found that it needs this addition. Fixing the flaw led to a fundamental shift in the way we understood nature. ecause of that, all of the E&M equations were renamed in his honor.
6 PES 1120 Spring 2014, Spendier Lecture 38/Page 6 So now we are ready to write down all 4 Maxwell's equations: Law Equation Physical Interpretation Gauss's law for E q Electric flux through a closed enc E da is proportional to the 0 charged enclosed. Faraday's law d E Changing magnetic flux ds produces an electric field. Gauss's law for The total magnetic flux through a closed is da 0 zero. (no magnetic monopoles) Ampere-Maxwell law de Electric current and changing ds 0Ienc 00 electric flux produces a magnetic field. In the absence of sources where, q enc = I enc = 0 (or in vacuum), the above equations become Law Equation Gauss's law for E E da 0 d Faraday's law E ds Gauss's law for da 0 d Ampere-Maxwell law ds 00 E An important consequence of Maxwell s equations is the prediction of the existence of electromagnetic waves that travel with speed of light 1 1 c T m / A C / N m / m s. The reason is due to the fact that a changing electric field produces a magnetic field and vice versa, and the coupling between the two fields leads to the generation of electromagnetic waves. The prediction was confirmed by H. Hertz in (need to use Stokes theorem and wave equation to proof...)
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