Question

How do you imagine you can increase the magnet field from a magnet (Faraday's Law)?

Answer 1

Faraday's Law: Gauss' law tells us that "charges create E fields". And we know that 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 = ?V (=E ?x) Michael Faraday, a British physicist 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 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 But before we can write it down, we need to first define one relevant quantity we haven't seen yet.Imagine a B field whose field lines "cut through" ("pierce") a loop. Define ? 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 ? = B? A = B A cos? B? is the component of B perpendicular to the loop: B? = B cos?. The UNIT of magnetic flux = [?] = T m^2 = Weber = Wb. If B is not uniform, we find the flux by adding it up over little patches of area, € ?B = ?? B? dA... Go back to Ch 24 where we defined electric flux, it's all pretty much the same! Examples of calculating magnetic flux: Here (picture to the left) ? = B A, because B is perpendicular to the area. (?=0) Here (picture to the right), ? =0, because B is parallel to the area. (?=90. ) No flux: the B field lines don't "pierce" this loop, they "skim" past it... Here, (picture to the left), ? = B A cos?. The flux is reduced a bit because it's not perfectly perpendicular. Just like Electric flux - it tells the number of B field lines "poking through" a small loop

Faraday’s Law: The induced EMF in any loop is EMF = - d?/ dt . (? is magnetic flux, t is time, this is the rate of change of flux through an area) • 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. • EMF is a lot like the battery's voltage, except it's not "localized", it's distributed around the loop we're considering. If you want a formal definition of EMF, it's € EMF = ? E? dL, which looks a lot like € ?V = ? ? E? dL • 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 Wb sec = Tm2 sec = N Am ? ? ? ? ? ? m2 sec = Nm Asec = J C= 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=-Nd?/dt. (Do you see why?) • Since ? = B A cos?, 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, ?=0, as shown. (Remember, ? 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|=|??/?t| = [ (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 (or perhaps warm up the wire of the loop). 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.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). • 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) 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, and so 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 induce an upwards B field, to try to keep things as they were. B(induced) may be small: it probably won’t succeed, but it tries. The direction of induced current is shown to the left. B(induced) points up, opposite the change in B(ext). (Here, this just happens to be the same direction as B(ext) was originally, but that’s irrelevant, it's a coincidence here.) If B(ext) instead starts to increase with time, then to fight that change you will induce a downward B,You should look at HRW Fig 31-5 and try to figure out which way i should go. It takes practice to get Lenz’s law.) Example: A metal bar slides along conducting metal tracks in a uniform B field pointing into the page. Push the bar to the left (as shown), and consider the conducting loop consisting of rail + slider. The area inside that loop is increasing, and so flux through the loop (B*A) is also increasing. (A = L*x, and x is increasing with time) |EMF| = | d?/dt | = |B dA / dt | = B L dx / dt = B Lv (Because v = dx / dt is the speed of the sliding bar) That means current flows around the loop, by Faraday’s law. If you put a light bulb somewhere in that circuit, it’d glow. The bigger B is (or the faster you slide the rod), the more current. Now, what direction will the current flow, CW or CCW? Lenz’s law will tell us! The external flux is into the page and increasing with time. So the change in flux is into the page. (Do you see that?) Lenz’s law says current will start to flow to fight the change. That current will induce a new B that points out of the page. By RHR #1b, that means CCW. Note: It’s not that B(induced) points out of the page because B(ext) is into the page. That’s a coincidence. It’s opposite the CHANGE in flux, not oppose the direction of flux. E.g., If instead you push the slider to the right, B(ext) is of course the same, but now the flux is decreasing with time, that’s opposite: the B(induced) will also be opposite, i.e. the current flows CW!! HRW Fig 31-10 is similar. There is a totally different way, kind of “previous chapter style”, to reach the same conclusion about the direction of induced current in the previous example. Consider a small + test charge sitting somewhere in the slider. It (and of course every other atom, electron, etc. too ) moves along with the slider to the left, with velocity v. It sits in a uniform B field. That means it feels a force (F = q v B), and the direction is given by RHR #2, try it yourself, convince yourself it is DOWN. But it’s a test charge in a conductor - it’s free to move. What that means is the B field thus forces test charges down the slider, which means a current I down - exactly the direction we got before (from Lenz’s law) Cool - a rather different way of looking at it, but the same result. Induced Electric Fields: EMF, like any "voltage difference", tells you about work per charge, or € EMF = ? E? dL. Which means that we can rewrite Faraday's law as: € E? dL = ? d?B dt ? = – d dt( ?? B? dA) (This reminds me a little of Ampere's law, € B? dL = ?µ0 ? I = ?µ0 dq dt , something you might think about... but not for now) Faraday's law says "changing B (in time) creates E fields"! (Once you have E fields, they can drive currents, which is what we've been looking at this whole chapter) Example: Consider a solenoid which has a uniform B field inside. Suppose you increase the current through the solenoid, so B inside changes with time. Now look outside the solenoid, (e.g at point P, shown). where there is no B field at all, ever. Faraday's law says if you follow the dashed loop, and integrate E . dL, you will not get zero, you get € ? E? dL = – d dt( ?? B? dA) = ??a 2 dB dt . This says that although B is (always!) zero out there, the changing B field inside the solenoid creates a nonzero E field out there. The E field runs in "circles" around the solenoid. The direction will depend on whether B is increasing or decreasing. (In the case shown, if B is increasing, Lenz' law tells us that the E field points counterclockwise around the loop shownIn this last example, since it's totally symmetric, E must be the same all the way around the loop at some fixed radius r, which means we can do the "line integral" on the left side of the equation on the last page, to get E (2 ?r) = ? a^2 dB/dt. But recall B(solenoid) = µ0 n I (where n is the number of turns per unit length) so we have E (2 ?r) = ? a^2 µ0 n dI/dt, or E = (µ0 n a^2/ 2 r) dI/dt. (CCW in the case shown. If I was decreasing, it'd be CW) In general, E that arises from Gauss' law (i.e. from static charges) "diverges" from the charges, whereas E that arises from Faraday's law (i.e. from changing magnetic flux) tends to "run rings" around the changing flux. This makes for "non-conservative" forces, because if you were to allow a little charge to run around that ring, there would be a net work done on it. That is, the work done on the charge as it moves from place to place depends on the path it takes! It also means that, although "electrical potential" (which is potential energy/charge, or voltage) is perfectly well defined (and very useful!) for static charge configurations, it really isn't useful or even meaningful when you have time varying magnetic fields. That's why we've starting introducing the term EMF in this chapter.

Electric bell

Many objects around you contain electromagnets. They are found in electric motors and loudspeakers. Very large and powerful electromagnets are used as lifting magnets in scrap yards to pick up, then drop, old cars and other scrap iron and steel.

Electric bell

Electric bells like the ones used in most schools also contain an electromagnet.

1. When the current flows through the circuit, the electromagnet makes a magnetic field.

2. The electromagnet attracts the springy metal arm.

3. The arm hits the gong, which makes a sound.

4. The circuit is broken now the arm is out of position.

5. The electromagnet is turned off and the springy metal arm moves back.

6. The circuit is complete again.

The cycle repeats as long as the switch is closed. Check your understanding of this with the animation.

Heating effect

Household wires and cables transfer electricity very efficiently. That means they don’t lose much of the energy as heat on the way. The electricity flows freely from the plug and is transferred into other forms of energy in household appliances, like toasters, lights and CD players.

Heating elements

The heating element in a toaster is made from a special wire different to wire in the power cables. The wire in the heating element transfers lots of its electrical energy into thermal energy and gets very hot. Exactly what you need to make toast or run an electric heater.

Energy efficiency

In most uses of electricity we don't want to produce heat. Any heat that is produced is wasted energy. Appliances that transfer less of the electricity they use into wasted heat are more energy efficient.

For example energy efficient light bulbs transfer less of their energy into heat. They also use less energy than normal bulbs to produce the same amount of light. So the energy efficient light bulb wastes less energy.

Electrolysis

Electrical energy can cause chemical reactions. This happens in a chemical process called electrolysis.

Electrolysis of water

One use for electrolysis is to break down water (H₂0) into hydrogen (H₂) and oxygen (0₂).

Electrolysis of water and sulfuric acid solution

1. Pieces of metal, called electrodes, are placed in the water.

2. A little sulphuric acid is added to the water to make the process work.

3. The electrodes are connected to a battery.

4. The water completes the circuit between the electrodes and current flows.

Where the electrodes come into contact with the water chemical changes take place, producing bubbles of gas. Oxygen at one electrode and hydrogen at the other one.

This electrolysis produces twice as much hydrogen as oxygen. They are both useful gases used in many chemical processes in industry.