A changing magnetic field flux can create an

A changing magnetic field (flux) can create an emf (DV) Demo

Recalling magnetic flux Recall electric flux: General expression Uniform field, flat surface Magnetic flux is: General expression Uniform field, flat surface General Uniform field, flat surface

Recalling Gauss’s Law for magnetic flux As we have seen, magnetic forces come from electric charges in motion. There are no free magnetic charges. Magnetic field lines diverge from N poles and converge into S poles, but they do not begin or end at either pole. Then Qmagnetic = 0, so that there cannot be enclosed charge. Gauss’s Law for magnetism is then: We have encountered magnetic flux before, and this is one of its important properties. Still, this law does not have a lot of direct applications, but Faraday’s Law of Induction, introduced in this section, does!

Also recalling Faraday. (Cage, the Farad, etc. ) There’s more! From Wikipedia: Michael Faraday, (1791 – 1867) was an English chemist and physicist who contributed significantly to the fields of electromagnetism and electrochemistry. He established that magnetism could affect rays of light and that there was an underlying relationship between the two phenomena. Some historians of science refer to him as the best experimentalist in the history of science. It was largely due to his efforts that electricity became viable for use in technology. The SI unit of capacitance, the farad, is named after him. Faraday's law of induction states that a magnetic field changing in time creates a proportional electromotive force.

Faraday’s Law of Induction The Law: If the magnetic flux through a closed loop is changing with time, an emf is generated around the loop. More precisely: Flux passing through the loop Voltage drop, DV, around the loop (We’ll discuss the minus sign a bit later. ) If there are N turns of wire in the loop, each sees the same changing flux. Then: A circuit will respond to this emf the same way it would to a battery (with no internal resistance). In the loop at right, if the magnetic flux inside the loop is changing, then: emfloop = DV = RI. Discuss “where”. I R

Making magnetic flux through a loop change with time 1. Increase or decrease the strength of the magnetic field through stationary loop. 2. Move a non-uniform field source toward or away from the loop. 3. Move the loop in a non-uniform field. 4. In a uniform field, change the area of the loop. 5. All of the above, in any combination. When motion is involved, the result is called “motional emf”. (emo? )

Demo Basic alternating current (A. C. ) generator In a basic A. C. generator, a permanent magnet provides a reasonably uniform magnetic field. As the generator loop turns in this field, the flux through the loop changes sinusoidally with time. This causes the output voltage (induced emf) to also change sinusoidally, 90 degrees out of phase with the flux. We can see why this happens by applying Faraday’s Law of Induction to the flux: Let: Then: Motional emf

Demo Basic D. C. Generator Like the D. C. motor, the D. C. generator has a split-ring commutator. This swaps the direction of the induced emf whenever it would go negative. Note: Basic D. C. motors and generators have the same construction, and are interchangeable. As you can see, the basic generator produces a very “coarse”, wavy, D. C. output voltage. The output can be smoothed with more loops (eg. 3) and more splits in the commutor ring (eg. 6). It can smoothed further by the use of diodes.

More motional emf More realistic. Use to discuss energetics.

Lenz’s Law The direction of any magnetic induction effect is such as to oppose the cause of the effect. This may sound a bit mysterious, but it is actually an easy way to figure out the direction of induced magnetic fields and induced currents without having to go through a chain of right-hand rules.

Lenz’s Law: looking at cases

More generally: the sign of the induced emf

A solenoid inside a loop. If the outer loop is turned into a solenoid, we can understand the demonstration with nested solenoids.

The Faraday disk dynamo Calc

A loop entering, then leaving, a region of uniform magnetic field R Consider (1) the induced emf and current in the loop, (2) forces on the loop as it travels from -2 L to 2 L, (3) work required to maintain constant v, and energy flow. Calc

Eddy currents and damping Use analysis from previous slide to explain what is happening here.

Metal detection using eddy currents

Induced electric fields: Faraday’s Law in terms of fields Again, the Law: Faraday’s Law of Induction, as we’ve used it so far, looks like it’s meant to be applied to voltage drops only. But let’s analyze the emf around the loop in terms of electric field. Recall that DV along a path is the line integral of E. We can make this path into a loop, and rewrite the left-hand side as: I E R For example, in the circuit at right, we see that there is an electric field circulating the loop. Not path independent any more! Try both directions. Now imagine erasing the circuit and using a “calculational loop”. This equation also applies to free space, and says that a changing magnetic field can create an electric field. This is “half” of what’s needed for electromagnetic waves!

Staying on a theoretical roll… We’ll be getting back to circuits, and a new circuit element, the “inductor”, in a few slides. But we’re at the point where, with one modification of Ampere’s Law, we will have all four of Maxwell’s Equations in hand. These, together with the Lorenz Force Law, completely specify electromagnetism. So we’ll do this first.

James Clerk Maxwell, 1831 -1879 Scottish mathematician and theoretical physicist. His most significant achievement was formulating a set of equations that for the first time expressed the basic laws of electricity and magnetism in a unified fashion. He also developed the Maxwell distribution, a statistical means to describe aspects of the kinetic theory of gases. These two discoveries helped usher in the era of modern physics, laying the foundation for future work in such fields as special relativity and quantum mechanics. “[The work of Maxwell]. . . [is] the most profound and the most fruitful that physics has experienced since the time of Newton. ”—Albert Einstein, The Sunday Post

Maxwell’s addition to Ampere’s Law Recall Ampere’s Law: As Maxwell did, we consider a capacitor being charge by a current i. C. First, construct an Amperean loop around the wire and notice that we can find the B field around the loop from the current passing through the plane surface. But, mathematics allows us to distort the Amperean surface membrane until it passes through the gap of the capacitor. In this region, there is no current flowing, but instead there is a changing electric field. Maxwell realized that the second surface must give the same answer, and added a term to Ampere’s Law. Including this new term, we get the Ampere-Maxwell Law: The charge on the capacitor is: “Current” through the capacitor is then:

Maxwell’s Equations Gauss’s Law for E Gauss’s Law for B Faraday’s Law of Induction Ampere-Maxwell Law Notice, from the 3 rd and 4 th equations, that changing magnetic flux can create electric fields and changing electric flux can create magnetic fields. This can happen even when there are no charges or currents present, and leads to the creation of electromagnetic waves light!

Magnetic field inside a charging capacitor Find