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!

Though free to think and act, we are held together, like the stars in the firmament, with ties inseparable. These ties cannot be seen, but we can feel them.

—
Nikola Tesla

Melting aluminum with an electromagnet.

Wave of the future: Terahertz chips a new way of seeing through matter

Electromagnetic pulses lasting one millionth of a millionth of a second may hold the key to advances in medical imaging, communications and drug development. But the pulses, called terahertz waves, have long required elaborate and expensive equipment to use.

Now, researchers at Princeton University have drastically shrunk much of that equipment: moving from a tabletop setup with lasers and mirrors to a pair of microchips small enough to fit on a fingertip.

**An Example of the Coordination of Magnetic Field Lines from a Bar Magnet.**

**Magnus, the Future of Science.**

Pictures like this is what hook people to astronomy, right? It looks so majestic and magical, making it hard to believe that something this astounding can exist. Wouldn’t you just want to take a trip to space to see one with your own two eyes?

As many people aren’t aware, if you had the ability to travel to one of these nebulae, it would look different. You may not be able to see it.

Why is that so?

In the universe, there exists an electromagnetic spectrum. This spectrum consists of the following (highest energy to lowest energy): gamma rays, x-rays, UV rays, visible light, infrared waves, microwaves and radio waves.

The human eye can only detect colours that are in the “visible light” range. This includes: red, orange, yellow, green, blue, indigo and violet (aka the colours of the rainbow) and different variations of those colours.

Astronomical objects, such as the Rosette Nebula pictured above (located 5,000 light years from the Earth), give off different forms of electromagnetic radiation. These forms include gamma rays and radio waves. But they give off, if any, very little amounts of visible light. Since our eyes can only detect visible light, we wouldn’t be able to see them very well and their colours wouldn’t be as defined.

Astronomers have to do some editing in order for the pictures to properly turn out how they do. A telescope will take the image with different filters, which will allow different wavelengths (different parts of the electromagnetic spectrum) of light to be seen (the higher the energy, the smaller the wavelength). Once pictures with different wavelengths were taken, the images will be overlapped, the filters combining, to make the amazing images we see today.

So maybe it would be better to just sit and stare admiringly at pictures of nebula on your laptop screen. Too bad we couldn’t see all wavelengths of the electromagnetic spectrum…..

Image Credit & Copyright: Arno Rottal (Far-Light-Photography)

“So he built this device and waited. The device wasn’t perfect, and occasionally one of the loops would send a signal, and on even rarer occasions, two loops would send a signal at once. But you’d need eight (and exactly eight) for it to be a magnetic monopole. The apparatus never detected three or more. This experiment ran for some months with no success, and was eventually relegated to being checked up on only a few times a day. In February of 1982, he didn’t come in on Valentine’s Day. When he came back to the office on the 15th, he surprisingly found that the computer and the device had recorded exactly eight magnetons on February 14th, 1982.”

The laws of electromagnetism could have been incredibly different. Our Universe has two types of electric charge (positive and negative) and could have had two types of magnetic pole (north and south), but only the electric charges exist in our Universe. At a fundamental level, between electricity and magnetism, nature is not symmetric. But it could have been! Magnetic charges could move and make currents; a changing electric field could induce them; north and south poles could be separated an infinite distance. Magnetic charge could even have been a fundamental property of black holes. In 1982, Blas Cabrera announced the first detection of a long-sought-after magnetic monopole event, and the physics world went crazy. But nearly 35 years later, a second monopole has never been found. Despite our dreams, it looks like nature isn’t symmetric after all.

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’.)

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 Φ_{B} 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.

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 2*π* radians, allowing us to define the angular frequency *ω* = 2*πf*. 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* = 2*πf* 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*) = 2*π* = 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*) = 2*π*

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.

So,

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!

I didn’t think it was possible, but the Nerd Herd managed to out-nerd itself today.

It started with fuck, marry, kill: Newton, Gauss, or Einstein. This then lead to fuck, marry, kill:

Div, grad, or curl (“the Laplacian would be an extramarital affair,” and “the D'Alembertian would be that one night in Tijuana”)

Sin, cos, or tan

e, pi, or the square root of two

C, C++, or Fortran

Diffy Q I, Matrix I, or Calc III

u substitution, trig substitution, or integration by parts

quantum, electricity and magnetism, or mechanics

the Laplacian, the Hamiltonian, or the Lagrangian

Gauss’s laws, Faraday’s law, or Ampere’s law

…We tallied all this up on the whiteboard, and we’re gonna put the results in a spreadsheet and then analyze it later.

Also, if they were the Golden Girls, the gradient would be Sophia, Dorothy would be the Laplacian (“because she came from Sophia”), curl would be Rose (“because she’s weird”), and divergence would be Blanche. Alternately, John would be curl, Paul would be the Laplacian, George would be divergence, and Ringo would be the gradient.

And there you have it.

The Human body is composed of over 60 Trillion individual cells, each containing within itself a perfect and complete version of itself ~Luce Cannon

The change of sky colour at sunset (red nearest the sun, blue furthest away) is caused by Rayleigh scattering by atmospheric gas particles which are much smaller than the wavelengths of visible light. The grey/white colour of the clouds is caused by Mie scattering by water droplets which are of a comparable size to the wavelengths of visible light.

Electromagnetism

*See previous posts about Coulomb’s Law*

Coulomb’s law describes the force **F** between two stationary charges *Q *and *q *separated by a displacement **r**. It is given as

where **r** with affixed hat is the unitary vector of **r** (*i.e*. the vector providing only the direction of **r**) and* ϵ*_{0} is the permittivity of free space (*ϵ*_{0} ≈ 8.854 187 8F m^{-1}).

Often, Coulomb’s law is written in terms of the Coulomb constant, *k*_{C} defined such that

This definition helps simplify a lot of results found in electrostatics and simplifies Coulomb’s law to

though this is a result which we shall not use.

For a system of *N *discrete charges *q*_{i} located at a displacement **r**_{i}, we can apply the superposition principle, which mathematically states that

where **F**_{i} is the Coulomb force due to the *i*^{th} charge, to determine the total force on a charge *Q* from each individual charge *q*_{i} with which it interacts. Mathematically, this is

In turn, we can express this for a continuum of charges (such as a charged plate in which the charges cannot be individually defined). Instead we consider an infinitesimally small area of charge d*q *and express **F** in terms of the displacement **r** of this charge distribution from a point charge *Q*.

If *q *is the total charge within the distribution then

Often, charge distributions may also be more conveniently expressed in terms of the charge density of a particular contour, surface or volume (referred to collectively as the “domain”).

So, our element of charge d*q* is placed within a domain Ω of total charge *q*, Thus, if we define our spatially-dependent (*i.e.* non-constant) charge density λ(**r**) as the charge per unit length/area/volume (depending on the type of domain), then

and so,

In many cases, we may have a constant charge density throughout our domain, which implies *λ*(**r**) = constant = *λ*, and so

Although the force exerted between particles tells us a lot about an electrostatic system, it also helps up define something a little more interesting − the electric field, which we’ll examine more closely in the next post!

More Than You Ever Wanted to Know About Electrical Engineering: More About Inductance

We’ve been talking about three-phase circuits of the kind you might see in residential systems, generally at 120 or 208 VAC. The voltage you actually see on a utility line is much higher than this, though, as it is more efficient (and consequently cheaper) to transmit over long distances at very high voltages. So how do you go from the very high voltage circuits on a utility line to the much lower voltages in a commercial or residential electrical system?

The device to handle this problem is a transformer. To understand how it works, we’re going to need to take a closer look at inductors and a little bit of electromagnetic theory.

At this point, we know that an inductor is basically a coiled bit of wire that stores and releases energy in a magnetic field. Let’s examine how that works a bit more closely.

There are two physical laws that will be relevant for us here. The first is Ampere’s Law, which simply says that any time you have an electric current, you get a magnetic field. So if we pass a current through a coil of wire, we’ll produce a magnetic flux, φ.

The geometry of the coils actually amplifies this effect - the coils in close proximity to each other “catch” more of the overall magnetic flux. You’ll see a term called flux linkage, λ, get thrown around. This is just a measure of how much magnetic flux is caught up in the coil, and it’s directly proportional to both the number of turns of the coil, N, and the overall flux.

We’ve already said that the presence of the magnetic flux is related to the current flowing through the coil, so we should also expect that the flux linkage is proportional to current. The constant of proportionality between the two is the coil’s inductance. This is all inductance really is: a measure of the relationship between the current flowing through a coil and the magnetic flux it generates.

The other physical law of interest to us here is Faraday’s Law, which states that anytime we have a changing magnetic flux, it creates a voltage.

We can substitute in our equation linking flux linkage, inductance, and current here. Applying the chain rule, we get something that starts to look a little familiar.

In most circumstances, inductance is a constant. We don’t have to worry about it changing over time, so for our purposes, this equation reduces to the equation relating voltage, current, and inductance which we’ve used before.

**Young Science Lover Garrett Discovers a Way to Line Up Disk Magnets So When the First One is Rotated the Rest Fall In Place.**