# Bar magnet dropped through coil

If you have drop a bar magnet through a coil so that it goes all the way through I was told the graph of emf induced in the coil vs time looks something like this:

(emf induced is on y axis, time is on x axis)

The area of the pink part and the blue part are equal (sorry that they don't look equal in my diagram)

I understand why the blue part looks more stretched than the pink part -- the magnet is accelerating. What I don't understand is the direction of the blue part -- it is opposite (negative) to the pink part (positive).

Why does the s-pole induce an emf in the opposite direction to the one induced by the n-pole when a bar magnet falls through a coil? The field lines at the s-pole and n-pole are pointing in the same direction, so the induced emf should continue increasing in the pink part (rather than turning and going below zero) because the field from the n-pole is complemented by the field from the s-pole which is cutting the flux in the same direction as the one from the n-pole.

Thanks!

Note: Layman terms means you can use stuff from high school physics (left/right hand rules, lenz's law, etc.), but not e.g. calculus.

• Perhaps this image could help: hyperphysics.phy-astr.gsu.edu/hbase/electric/imgele/lenz.gif suppose north pole is down when releasing the magnet, then before reaching the coil situation is like upper left figure (counterclockwise emf) and after passing it it's like upper right (clockwise emf). If south pole is down it'll be bottom right and bottom left respectively. – Azad May 6 '15 at 9:39

One of Maxwell's equations is $$\nabla \times \vec{E} = - \frac{d\vec{B}}{dt} \, .$$ Consider an imaginary disk whose normal vector is parallel to the axis of the coil and which is inside the coil. If you integrate this equation over the area of that disk you get $^{[a]}$ $$\mathcal{E} = - \dot{\Phi}$$ where $\Phi$ is the flux threading the disk and $\mathcal{E}$ is the EMF drop around the loop. This is called Faraday's law.

So, for each imaginary disk inside your coil we get some EMF as the flux through that disk changes in time.

Now think about the bar magnet's descent. Suppose we drop it starting way above the entrance to the coil. It's far away, so there's no flux and no EMF. As it descends and gets close to the entrance of the coil, some of the magnetic field from the magnet threads the top few imaginary disks of the coil. The time changing flux induces some EMF. This is the initial rise in the red part of the diagram. As the magnet continues to fall, and enters the coil, more of its magnetic field is threading imaginary disks in the coil, so as it moves the time rate of change of total flux increases, so the EMF goes up. Note that the field lines above and below the bar magnet point in the same direction.

At some point, the bar reaches the middle of the coil. At this point, the amount of flux added to the top half of the coil by a small motion of the magnet is equal to the amount of flux removed from the bottom half. Therefore, at this point the EMF is zero. This is the midopint of the diagram where the EMF crosses the horizontal axis. The falling part of the red section is just the approach to the mid section of the coil.

As the bar magnet exits the coil, more flux is leaving than is entering, so the EMF versus time in the blue section is just the opposite (except for the stretching which you already understand) of the red section.

[a]: On the right hand side, the area integral of the magnetic field is the flux $\Phi$ by definition, and the time derivative just goes along for the ride. On the left hand side you are doing an area integral of a curl of a vector, which by Stokes's theorem is equivalent to the line integral of the vector itself around the boundary of the area. The line integral of the electric field vector is the EMF by definition.

• OMG, I actually understand this! There's still one thing bothering me just a tiny bit: When the magnet is exiting the coil, why is "more flux leaving than is entering"? I was thinking something like this: We can think of the n-pole producing half of the total flux from the magnet, and the s-pole producing the other half. So as soon as the magnet passes the point where the flux cancels out (this is the part where it transitions from pink to blue in the diagram), the total flux going out is more than half of the total flux of the magnet, meaning that more is going out than coming in. – user45220 May 8 '15 at 16:21
• I don't entirely understand what you're trying to say in the comment. The magnetic field lines extend some distance away from the magnet. As the magnet nears the bottom of the coil, the field lines below the magnet start poking out the bottom of the coil, so the total amount of flux changing inside the coil goes down. Does that make sense? – DanielSank May 8 '15 at 19:30
• Wait, it's about how much there is inside the coil, I was just over-complicating it in my last comment. – user45220 May 8 '15 at 19:33
• Also this answer is the one I understood most so it's the one I'll accept (in ~ 20 hours), good job – user45220 May 8 '15 at 19:34
• @user45220 really glad you understand it :D – DanielSank May 8 '15 at 19:51

Here are some pictures to explain:

For the case of constant velocity:

And for the case of accelerating (the case you are actually interested in):

And, one final note to sum it all up, the induced EMF is the coil is $$\left(-\frac{d\Phi}{dt}\right)$$. I chose my $$\vec a$$ (the area vector) pointing "to the right" which means that the positive sense of current is in the direction of the fingers when using the right hand rule with the thumb in direction of $$\vec a$$. Since the induced EMF is actually the negative of the values shown in the $$\frac{d\Phi}{dt}$$ drawings, the current flows in the opposite direction as the direction that the fingers on the right hand are pointing.

E.M.F(induced)= -(rate of change of magnetic flux).
Which implies that the direction of E.M.F depends upon the sign of the Rate of Change of magnetic flux.
When the magnet is moving into the coil, magnetic flux is increasing(Because magnetic feild(and thus flux) are greater nearer we get to the poles), while when it is coming out magnetic flux is decreasing.Therefore, when entering the loop, rate of change of magnetic flux is positive, and while coming out, it is negative.
Thus, EMF has two different signs.
Hopefully this solves your question in Layman's terms(this is a really simple answer) :)
I apologize for the poor formatting, I will format it soon. Also, I will explain in mathematical terms also soon.

• Is this Luck, I am the first answerer by just minutes! – Mathsman 100 May 8 '15 at 16:15
• Hello, thank you for your answer but I don't think it explains why the flux is decreasing when the magnet is coming out. You just stated it without explaining: "When the magnet is moving into the coil, magnetic flux is increasing, while when it is coming out magnetic flux is decreasing." – user45220 May 8 '15 at 16:23
• Hope this works – Mathsman 100 May 8 '15 at 16:52
• When the magnet it far away and moving towards the coil the flux is small, when it is close and moving towards the coil the flux is large. Thus, as it approaches the flux is increasing. When the magnet is close and moving away the flux is large, when it is far away and moving away the flux is small. Thus, as it moves away the flux decreases. The magnitude of the flux only depends on how close the magnet is to the coil, that's why moving towards --> increasing flux whereas moving away --> decreasing flux. – hft May 8 '15 at 17:50

Layman's Explanation:

Long ago the magnet was far far away and the magnetic flux through the coil was essentially zero.

If you measure the induced emf as a function of time with time as the x axis and the induced emf as the y axis then the area under the curve an above the x-axis between the vertical lines $t=t_1$ and the vertical line $t=t_2$ is the total change in magnetic flux from time $t_1$ to time $t_2$. This is a law of physics, known as Faraday's law. It is not explained in terms of anything more fundamental, and is motivated by its agreement with experimental observations.

Apply that to the previous paragraph and the area under the emf graph up to some line $t=T$ is the actual magnetic flux through the coil at time $t=T$.

So as the bar magnet falls the flux gets larger and larger until the magnetic is literally inside the loop. You can see this by drawing the magnetic field lines around the magnet and imagining different places a loop could be. If centered on the loop all the field lines come through the middle of the magnet and out the north pole and each one goes in a big loop before it turns around and comes back in through the south pole of the magnet. Each one has some distance away where it is fully pointing the opposite direction as when it goes from north to south. Those that turn around outside the coil contribute to the magnetic flux because the one going north contributed, and the same field line was outside the coil when it pointed south. Those that turn around within the loop go in and out and net don't contribute any flux (they contribute an equal amount positive and negative flux). If you pick a field line that pointed south outside of the coil (the ones that contributed on their way north but on their way south) and you imagine placing the coil farther up eventually the coil and the field line intersect if you put it even farther, the field line now fails to pierce the area enclosed by the coil as doesn't contribute any flux at all. So the flux goes down. Each field line had such a point, so the flux just goes down and down and down as the coil is farther from the magnet.

You can do the same thing by sliding the coil downwards but this time the field line crosses the plane of the coil outside the coil on its way to the south pole of the bar magnet. So the net effect is that the flux is weaker the farther the coil is from the magnet, and when the magnet was centered on the coil the flux contained all the field lines that didn't turn around within the coil (it had those but it had them coming and going so no net effect). And it only gets weaker than that.

So when the coil is farther away from the magnet you have less flux in the area enclosed by the coil. So we expect the area under the emf curve to be zero way to the left, and then as you imagine moving a vertical line for $t=T$ to the right you start getting more and more area to the left of the line $t=T$ until the area equals the area magnetic flux when the coil is centered on the magnet. That's when the flux is biggest. At this point the flux has to get smaller as the magnet continues to move. We can make the area smaller by saying the area below the line $y=0$ Volts and above the emf curve counts as negative area. So we need to get that, and eventually when the bar magnet is far away the total flux is zero again so the negative area between the emf curve and the x axis (the area under the x-axis and above the emf curve) needs to equal the positive area between the emf curve and the x axis (the area under the x-emf curve and above the x-axis) so the total area is zero, to have the magnetic flux be zero. As it needs to be at late times, i.e. far to the right.

All we have left to explain is why the area to the right is more squished. Since the magnet is falling, it speeds up and fall faster and faster, so it moves away from the center fast later on so the flux changes faster later on. So we need to accumulate more area more quickly so we need a larger emf, a larger emf gives more area in a certain time, for instance the strip of area between the vertical lines $t=t_1$ and $t=t_2$ contains more area when $t_2-t_1$=1sec (in one second) if the emf is larger (and positive area if the emf is positive and negative area if the emf is negative). So the last part of the curve needs to look like a squished version of the first part, but even more squished nearest to the right.

As far as I can tell this is the first answer that doesn't talk about rates of change, and explains the why of each step in layman's terms.

Technically if you want to know which direction the emf goes you need to pay attention to whether you drop the magnet with the north pole up or the north pole down. No ideas change, the same kind of emf curves happen either way.

Why does the s-pole induce an emf in the opposite direction to the one induced by the n-pole when a bar magnet falls through a coil?

I'm sure that a complete description in 'layman terms' is possible, for this we haqve to talk about basics of EM induction.

You use a metallic wire which is bended to a coil. Inside the wire are available valence electrons. Every electron has three relevant features: electric charge, magnetic dipole moment and intrinsic spin. The amount of the value of these features are constant numbers. The electric charge is a constant number (in rest) and equal distributed in space.

The electrons intrinsic spin and magnetic dipole moment always are pointing together in the same direction (direction and sign!) but in a chaotic system (not influenced from outer forces and not 'frozen' like in permanent magnets) all directions are equal distributed in space. I don't know, was you told explicitly that only electric charges attract / repel each other and only magnetic dipole moments do the same to each other. Never a electric charge interact with magnetic dipole moment directly.

Electromagnetic induction is the magic thing we use to generate current, to make strong electromagnets and to use electric drives (Lorentz force). The mechanism behind this phenomena is based on the relative movement of magnetic and electric field to each other. Imagine an electron in space, like in the sketches in this question, and bring a magnet close to this electron. The magnet will align the electrons magnetic dipole moment and the intrinsic spin too.

Did you ever try to revolve a rotating bicycle wheel? The wheel works against such movement. The same happens with free distributed electrons then they get under the influence of a magnetic field. In the coil we observe a current. Imagine for just a second that in electrons the spin would be aligned to the magnetic moment parallel and anti-parallel. That would be a catastrophic case for our modern live. In the wire to currents would be start to flow which led to the stop of this currents immediately du to an area with no valence electrons. So let as be happy how nature exists.

We have to start the last step in our journey of EM induction. The falling permanent magnet alignes the electrons intrinsic spin all in one direction and a current starts. Thinking precisely this process has to work only once. Once aligned nothing more could happens. Fortunately there is one phenomena more. It was found (hundred years ago) that any acceleration of a body is escorted by electromagnetic radiation. When a electron get aligned and moves in a circle this electron emits photon(s). This let to a misalignment of the magnetic dipole moment and the game starts again. It is important to understand that such mechanism works only under the circumstances of relative movement of the involved components.

Now it has to be clear, why the induced current switches his direction when the permanent magnet influence first with the one pole and then with the other pole on the coil. The aligned intrinsic spins switches from one direction to the other and the current do so.

I'm not sure if a complete description in 'layman terms' is possible, but I'll have a go at an answer without advanced mathematics.

We will need to use some equations: in particular, Faraday's law that the EMF is given by the rate of change of the magnetic flux through the coil: $$\mathcal{E} = -\frac{\partial \Phi_B}{\partial t}$$ This means that the integral of $\mathcal{E}$ over time effectively gives the change in magnetic flux: $$\int \mathcal{E} dt = -\Delta\Phi_B$$ The area of the two parts must then be equal, as the magnetic flux built up from zero when the magnet enters is the same as the magnetic flux that goes to zero when the magnet leaves. This is something that cannot be understood without calculus.

As for the direction of the EMF, (and we don't need to use calculus here, to address your main question), notice the statement of Lenz's law is that:

If an induced current flows, its direction is always such that it will oppose the change which produced it.

The direction of current flow in the loop is clearly given by the direction of the EMF. Now, as you point out, the field lines are pointing the same way at both the North and South poles (we will assume that the north pole goes through first). But whereas the North pole enters the coil (when it is nearer to the coil), the South pole leaves the coil after the North pole crosses it, and this makes all the difference. So, in the same direction, the strength of the magnetic field/flux through the coil is increasing when the north pole is coming in, but decreasing when the south pole is leaving.

In the former case, there is one direction of current to decrease the increasing flux, in the latter case, there is another direction of current to increase the decreasing flux. Thus, the directions of the EMFs between the two cases are opposite.

According to Lenz's Law, an EMF generated by changing magnetic flux in a coil creates an electric current that flows in the direction which generates an induced magnetic field opposed to the changing flux that produces it.

So, for example, as the north pole of a magnet passes through a coil, current travels counter-clockwise in the coil (looking in the direction of the magnet's passage). This creates an induced EMF with polarity that acts against the increasing flux of the magnet.

By Lenz's law, the induced magnetic field always acts so as to try to keep the magnetic flux inside the coil constant. As the north pole of the magnet enters farther into the coil, the induced EMF tries to COUNTER the increasing flux inside the coil created by the passage of the magnet. It gets stronger, with polarity opposite to the passing magnet.

When the north pole exits, and when the south pole of the magnet enters the coil, the induced EMF begins to AUGMENT the passing magnet's EMF so as to try to keep the decreasing flux constant. In order to augment the decreasing flux in the coil, the induced EMF must have the same polarity as the passing magnet.

The change in polarity explains why the curve is above the x axis until the magnet begins to exit the coil, when it shifts to below the x axis.

Here is a diagram that may help: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html#c2

This may help, too (scroll down to the 3rd page where Lenz's law begins): http://spiff.rit.edu/classes/phys213/lectures/lenz/lenz_long.html

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