21p-s  asked:

hey! i'm doing electric power at school and i'm really confused about which equation to use when you're finding the induced emf? (-n∆flux) / (∆t) or nBA2pif???? thank you!

Hi! Both equations you mention are actually very valid to help you find the induced EMF! Since I am in the midst of making posts about electromagnetism, I’ll go through this is detail.

Although I will use both calculus with vectors and scalar, pre-calc representations, for your level I’d recommend the equations with non-infinitesimal changes (signified by an ‘approximately equal to’ ‘≈’ symbol and a ‘Δ’ symbol in place of ‘d’.)

The first equation

If we consider the first equation for induced EMF you’ve provided, which states that

(equation 1) where N is the total number of turns in a loop of wire, t is time and ΦB is the flux of an inductive magnetic field B through a surface S (which we can define for each situation,) given by

in which: “⋅” represents the scalar (‘dot’) product; ϕ is the angle between the field lines of the magnetic field and the normal to the surface. It is worth noting that the cos ϕ part of the expression comes directly from the scalar representation of the dot product.

This is the most general expression for the induced EMF – describing the effect a changing magnetic field (or a moving/changing surface, for that matter) has on the motion of charges through a circuit. It tells us that when we move a conducting rod through a loop of current-carrying wire, we expect to observe a current induced through the rod since lines of flux are being intersected by our rod (which is our surface, in this case.)

However, if we move our rod through the wire loop—which will, for now, be carrying a DC (i.e. direct current; non-changing) signal—and hold it at the centre, we notice this new current produced by the induced EMF will drop off in proportion with how fast we move it through the field. This is because less lines of flux are being intersected per unit time, signified by the ratio of flux change to time change. This is an important feature of electromagnetic induction because it means that if the current in the first wire is constantly changing, we need not vary the position or dimensions of our rod to allow an induced current.

The second equation

A changing field is produced using an AC circuit, which varies the voltage from a positive value to a negative value within a specific time frame, given by the frequency f of the signal which oscillates over 2π radians, allowing us to define the angular frequency ω = 2πf. This principle is vital to modern life and has applications in communications methods such as Wifi, radar, GPS and Blutooth. I hope to write a set of posts about radio communications in the future but we’ll have to wait and see whether I end up having time!

We can visualise the inductive effects of an alternating current by considering how it is principally generated – using a rotating magnet in a solenoid of N turns (a solenoid is long of wire coiled multiple times to form a sort of wire tube.)

Let’s define our flux surface as the area of a wire loop so we can see the how the flux of the magnetic field through the solenoid. To simplify the situation, we’ll ignore the integration and say that S := A, where A is the cross-sectional area of the solenoid, which means that

It is worth noting the time-dependency of the angle here: our magnetic field source is now rotating, so the angle between the field lines and the surface normal will be changing with time, too! Thus, with a little bit of thinking, we can determine ϕ(t).

We know our magnet, which is producing field lines through its poles, is rotating with an angular frequency ω = 2πf and we know that f = 1∕T, where T is the time period, so

This may help us conceptually when considering the timing of the magnet’s angle, since we can express the time in terms of the total period of the motion.

Let’s say at time t = 0, the magnet has its North pole towards the surface; the angle here is zero, meaning the induced EMF is at a maximum.

One complete revolution later, where t = T, the magnet has returned to its original position and now has ϕ(T) = 2π = 0. Therefore, by substituting the current time for our period we find that

Through rearrangement (multiply both sides by t,) this becomes

and, since we know that ϕ(T) = 2π

and we know that ϕ(T) = ϕ(t), so finally:

This is our time-dependent angle, discovered through inference alone! We can check this for any stage rotation in terms of the period and will find the correct angle. This expression throughout in oscillations throughtout physics.


Now we have to use a little calculus. Recall our original expression for the induced EMF, given by equation 1, and substitute our new expression for the magnetic flux density:

By applying the chain rule, which states that

we find that

and so

which, when evaluated at t = 0 (or t = T, and so on) for maximum EMF, yields the equation you provided:

From following the derivation through, it can be seen that this expression outlines the maximum EMF induced in a wire which follows a regular looping pattern around a cross sectional area A (as with a solenoid of N turns) by a magnetic field density B for some alternating current (or regular oscillation) having a frequency f. This could be used for any AC-based induction, any system involving periodically rotating magnets or something involving periodic changes in solenoid dimensions. The key point here of that this equation requires periodic oscillation oscillation and, evidently, this describes a much more specific scenario than the first equation.

The mathematical derivation itself may be a little difficult to follow as it uses mathematics that you haven’t explicitly met yet but the key concepts lie in the derivation. Just try to understand the scenario outlined and the arguments made so you can know whether a scenario is applicable.

For a little more information, which should be more aimed at the specific syllabus you are studying, see the following posts:

I hope this helps! Let me know if you have any more questions!

Today in history: Michael Faraday

Michael Faraday was one of the most influential scientists in history, despite receiving almost no formal education. He inspired Albert Einstein, who kept a picture of Faraday on his study wall, and physicist Ernest Rutherford described him as ‘one of the greatest scientific discoverers of all time’.

Faraday educated himself, aged 14, reading books while being apprenticed to a local bookbinder. He went on to contribute to electromagnetic induction, diamagnetism and electrolysis, as well as discovering benzene, investigating the clathrate hydrate of chlorine, inventing an early form of the Bunsen burner and the system of oxidation numbers, and popularising terminology such as anode, cathode, electrode and ion, and establishing a reputation as the outstanding scientific lecturer of his time. Ultimately, his work enabled the development of electricity. 


[ 23/06/16 — 11/100 days of productivity ]
Today I finished writing my notes for integration, then went on to do notes for electromagnetic induction. I also found time to put up my photos (which I took down when my house was repainted).
Just 4 days left till school re-opens 😣

Pickups are the strips you see under the strings on an electric guitar. They turn the physical vibration of the strings into an electrical signal. On an acoustic guitar, you hear the sound waves created by the strings’ vibration. On an electric guitar, you hear the electrical signal after it’s been run through an amplifier and comes out of a speaker. (Although the Clear Science art director couldn’t have an amp when he was a kid, so would put his ear on the horn of the guitar and hear the sound waves traveling through the solid guitar body.)

Guitar pickups work by electromagnetic induction, so let’s talk about what that is. If you have a coil of wire (a copper coil is illustrated above), a magnet passing through the coil will induce an electromotive force (EMF) in the coil. This EMF will cause an electric current to flow. (Incidentally, this is also how electric generators work.)

This concept is called Faraday’s law of induction, discovered by Michael Faraday. The EMF is equal to the time rate of change of the magnetic flux through the coil. Magnetic flux means the amount of magnetic field per area per time. Said plainer, this means changes in the magnetic field will cause some kind of current.

We talked about guitar pickups and how they work. It has to do with electromagnetic induction, which says that when there’s a change in the magnetic flux through a coil of wire a current will be induced in the wire.

The little disks under an electric guitar’s strings are magnets. Each pickup has a magnet under each of the strings. (Just for the record, there are different kinds of pickups that don’t look like this, but what we’re describing is the most common kind.) Inside the pickup, there’s a wire coil around the magnets. This wire coil is connected (after going through the tone and volume knobs) to the output socket, which you connect to the amp.

Since the strings are steel, each string becomes magnetized due to the permanent magnet under it. When you pluck the string it oscillates from its original position. Since it’s magnetized, the magnetic field through the wire coil involves both the string and the magnet, and so the string’s oscillation changes the magnetic field inside the coil. (Up above we’ve represented the magnetic field with a blue shape, rather than drawing field lines.) These changes in the magnetic flux cause electrical current to go back and forth in the wire. This signal ends up at the amp and speaker, where it becomes sound waves.

Ever wondered why you don’t have to plug in an electric guitar? It makes its own electricity. 


World’s First Electric Generator

“Michael Faraday created the first electric generator in 1831 using a coil of wire and a permanent magnet. When the magnet was moved relative to the coil, current was induced in the coil. A similar experiment can be performed with a copper tube and a magnet. Although copper is not magnetic, it is a conductor. As the magnet falls through the pipe, the magnetic field changes over different sections of the pipe. This induces swirling currents (called eddy currents), which create a magetic field that opposes the motion of the magnet. This means work must be done to move the magnet through the pipe. This work generates the electrical energy, which is then dissipated as thermal energy in the pipe. The same basic principle is used to generate electricity throughout the world: moving a magnet inside copper coils.”