21-1 (SJP, Phys 2020, Sp ‘91) Faraday's Law: A steady E field pushes charges around, makes currents flow. We've used the word "EMF" for this occasionally, an EMF is any voltage difference capable of generating electric currents. Think of EMF = ΔV (=E Δx, remember that relation between V and E?) (Note: batteries have an EMF, but resistors do NOT. Even though an R can have a voltage difference across it, it is not generating it! Resistors don't make currents spontaneously flow, batteries can.) Michael Faraday, a British physicist (at the same time as Joseph Henry, an American, but Faraday published first) about 180 years ago discovered a remarkable new property of nature: Changing magnetic fields (not steady ones) can make EMF's. In other words, a time-varying B field can make currents flow. Imagine a wire loop sitting in a B field, like this: If the B field is steady then there is NO CURRENT, the bulb is dark. But, if the B field changes with time, the bulb lights up, a current flows through that wire (!) You might do this by e.g. just moving a big magnet closer, or farther away (yes, weakening the B field is still a change)... or move the coil itself closer (or farther) from the magnet face. There's no battery here, no external voltage source, but the bulb still glows! This effect is surprising, it's something new... Faraday spent only 10 days of (intensive) work on these experiments, but they changed the world radically. This is how most of modern society's electricity is now generated! Faraday worked out an equation (Faraday's Law) which quantifies the effect (how much current do you get?) But before we can write it down, we need to first define one relevant quantity we haven't seen yet. wire x x x x x x x x x x x x x x x x Bulb x B21-2 (SJP, Phys 2020, Sp ‘91) Imagine a B field whose field lines "cut through" or "pierce" a loop. Define θ as the angle between B and the "normal" or "perpendicular" direction to the loop. We will now define a new quantity, the magnetic flux through the loop, as Magnetic Flux, or Φ = B⊥ A = B A cosθ B⊥ is the component of B perpendicular to the loop: B⊥ = B cosθ. The UNIT of magnetic flux = [Φ] = T m^2 = Weber = Wb. Flux is a useful concept, used for other quantities besides B, too. E.g. if you have solar panels, you want the flux of sunlight through the panel to be large. House #2 has poorly designed panels. Although the AREA of the panels is the exact same, and the sunshine brightness is the exact same, panel 2 is less useful: fewer light rays "pierce" the panel, there is less FLUX through that panel. Examples of calculating magnetic flux: Here (picture to the left) Φ = B A, because B is perpendicular to the area. (θ=0) Here (picture to the right), Φ =0, because B is parallel to the area. (θ=90. ) No flux: the B field lines don't "pierce" this loop at ALL, they "skim" past it... (That's zero flux!) Here, (picture to the left), Φ = B A cosθ. The flux is reduced a bit because it's not perfectly perpendicular. B The"normal" to the loop ! House Solar panel (Lots of flux) (Less flux) Solar panel 2, same area, diffferent tilt. 1 2 B Area, A B Area, A B ! B"21-3 (SJP, Phys 2020, Sp ‘91) Faraday’s Law: The induced EMF in any loop is EMF = - ΔΦ/ Δt . (Φ is magnetic flux, t is time.) • If you put a loop into a B field, and then change the flux through that loop over time, there will be an EMF (basically, a voltage difference) induced. Current flows, if you have a conducting loop. • The formula says it is only the change in flux through the loop that matters. A huge B field (lots of flux) does NOT make the EMF, it’s the change in B with time that does the trick. • This equation has not been derived - it’s just an experimental fact! • Units are Wbsec=T m2sec=NA m! " # $ % & m2sec=N mA sec=JC= Volt (yikes! It’s a mess, but it works out. The formula gives the correct units.) • If you were to “pile up” N loops on top of each other, the effective flux will be increased by a factor of N, the formula becomes EMF=-NΔΦ/Δt. (Do you see why?) • Since Φ = B A cosθ, you can change the flux in many ways: you could change B, or area, or the angle between B and the loop. Example: B is perp. to this loop, θ=0, as shown. (Remember, θ is the angle from the normal) The area is A= (0.1m)^2 = .01 m^2 Suppose B is 1 Tesla, as shown, and then you turn it off, taking a time of 2 seconds to do so... Faraday’s law says there will be an “induced EMF”, or voltage, around the loop, |EMF|=|ΔΦ/Δt| = [ (1 T * 0.01 m^2) cos(0) - 0 / (2 sec) = .005 V If you had N=1000 coils (loops) of wire, all stacked (coiled) up around that same perimeter, you’d get |EMF|=5 V, enough to light up a small bulb. But remember, you’d only have this voltage for those 2 seconds while B was changing! Once B reaches 0 (and presumably stays there), there is no more change, and so |EMF| goes back to 0. B N (=4) loops B10 cm10 cm21-4 (SJP, Phys 2020, Sp ‘91) What’s that minus sign about in Faraday’s law? Don’t plug it in blindly - it’s only there as a reminder, you must figure out the direction of the induced current flow, or voltage difference (the direction of the EMF) by Lenz’s Law: • Induced EMF tries to cause current to flow. If current flows, it will create a new (usually small) B field of its own, which we will call B(induced). (You’ll need to remember RHR #1b: how currents in a loop produce B fields) • I will call the original or “outside” field: B(external) The direction of B(induced) opposes the change in the original B. \Note: B(induced) does NOT necessarily oppose B(external)(!!) It is opposite the CHANGE of B(external) (or more accurately, the change of flux). B is a vector, you really have to think about the direction of the change of that vector.... Lenz’s law is a mouthful! It tells you the direction that the induced current will flow. Nature creates a B(induced) to fight the change. Example: Consider a B(ext) that is up, and pierces a wire loop, as shown. It might be caused by a big old magnet or something. If B(ext) stays constant, there is no change, no current spontaneously flows around the loop. If B(ext) starts to decrease, nature will try to fight that change. (Remember, if an “up vector” is decreasing, the change is DOWN) Lenz’s laws says a current will flow (or try to flow) to
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