# Why doesn't the magnetic field of a coil “cancel out”?

In a coil, we can see that the current moves right, then left, then right, then left, and so on as it travels down the coil.

According to the right-hand grip rule, isn't the magnetic field going in one direction (downwards towards me) when the current moves left, and going in another direction (downwards away from me) when the current moves right?

Why doesn't the magnetic field "cancel out" as such?

• Why don't you try using a coiled up wire and moving your hand along it to see if you're not messing up the results of the RHGripRule? Or you can simply do the same by moving your hand in a circle and seeing if your fingers ever point upwards. Do they?? – mikhailcazi Jan 19 '14 at 7:03
• @mikhailcazi, I'm not saying they ever point upwards. I'm saying why doesn't the "downwards towards me" field cancels out the "downwards away from me" field? – Pacerier Jan 19 '14 at 7:05
• They do! That's why you only have a net downward field inside the coil. – mikhailcazi Jan 19 '14 at 7:08
• @mikhailcazi, Why does the image shows a net "downwards towards me" field then? (look at the 5 arrows indicating the magnetic field) – Pacerier Jan 19 '14 at 7:09
• I see what's troubling you. That isn't a downward towards us field. It's a 2d image, you can't expect it to represent a 3d configuration very well. See this (you'll have to go forward a bit): youtube.com/watch?v=V-M07N4a6-Y . If you look at it from the top, you will see field lines coming in radially. Not from only one side. And the arrows at the bottom are not towards us. They also go out of the magnetic field radially. It's just difficult to show because it's a 2d image trying to show 3d. – mikhailcazi Jan 19 '14 at 7:14

Let's go back and consider the Biot-Savart Law for a filamentary current:

$$B(r) \propto \int_C I(r') \times \frac{r-r'}{4\pi |r-r'|^3} \, dr'$$

As I've written it, $r'$ is some position on the coil (we integrate over the coil to get the whole magnetic field). $r$ is the position we want to find the field at--somewhere inside the coil's empty body.

Let's consider $r$ directly at the center of the coil. As we start integrating, we slide along the coil. I'll traverse the loop clockwise, so let's have $r'$ start at the top of the batter, come across the top of the page, and start traversing the topmost loop of the coil.

As we start on the rightmost part of the topmost loop, the instantaneous direction of current is out of the page. If we take $r = 0$, so that the center of the loop is the origin, then all we have is the vector $I(r') \times [-r'/|r'|^3]$ as the integrand. $-r'$ points inward toward the center of the coil. It should be clear that the resulting magnetic field from a small piece of the wire at this point in the coil is both downward and to the left.

Let's consider what happens when we get to the leftmost part of the topmost loop. The vector $r'$ is downward and to the left. The current is into the page. The resulting magnetic field is downward and to the right.

In general, as we traverse the topmost loop, each small piece of wire adds a magnetic field that is (a) downward and (b) pointing out of the coil. (I must remind that we're talking about the magnetic field only at the center of the whole coil right now).

If the topmost loop were rotationally symmetric, we could argue that any components that point away from the central axis of the coil must cancel. The real coil does not have this symmetry, but it's "pretty close" to being rotationally symmetric, and any such real components ought to be small.

All the other loops work basically the same way, contributing only net downward magnetic field when integrated over a whole circular loop.

For some reason, you referred to the magnetic field contribution from different points as being clockwise or anticlockwise. I do not understand this. This coil does not create any kind of closed magnetic loops that can be seen on this scale. The field is more clearly described using fixed directions (into or out of the page, left or right, down or up).

I think it's this reason that you thought "clockwise and anticlockwise should cancel". But you described it more correctly in saying that at both points, there is a downward component; the only thing that can cancel a downward component is an upward component! You were right to say that at points where the current moves left, the field's direction is down and out of the page, and that when the the current moves right the field direction is down and into the page. It's just that only the into/out of page components cancel, and net downward is left behind.

• If the into/out forces cancel, why does the textbook above show that the net resultant force is "down and out" instead of "straight down"? – Pacerier Jan 19 '14 at 5:28
• As I said, I carried out this analysis for the exact center of the coil. Going off axis (but still halfway between the top and the bottom) or taking an arbitrary point within the coil requires a separate (but similar) analysis. – Muphrid Jan 19 '14 at 5:35
• Do you mean that the net resultant force of the entire coil is "down and out" or "straight down"? – Pacerier Jan 19 '14 at 5:42
• Net resultant force of the entire coil? This term has no meaning. The Biot-Savart law tells us about the net magnetic field at specific points within the coil. That is all it can tell us. – Muphrid Jan 19 '14 at 5:43
• This new image shows the subsection of the loop: i.stack.imgur.com/xmcgt.png . The image shows the direction of the magnetic field as downwards and towards us when the current goes to the left (bottom part of the wire colored blue by me). I can understand that. But why does it show the magnetic field has the same direction (downwards and towards us) when the current goes to the right at the top part (colored red) of the wire ? – Pacerier Jan 19 '14 at 6:19

I guess pictures will help you to visualize this:

In your red region, just look at one point... the current is to the right and the by using the right hand grip rule, the direction of magnetic field will be anti clockwise

In your blue region, just look at one point... the current is to the left and the by using the right hand grip rule, the direction of magnetic field will be clockwise

• Exactly what I'm trying to say. I can understand the arrows for the blue section. But look at i.stack.imgur.com/MZbK5.png ... Isn't the image contradicting us for the red section? – Pacerier Jan 19 '14 at 7:02
• @Pacerier: How is it contradicting... its just that you are not shown the back side of the diagram.nothingnerdy.wikispaces.com/file/view/coil_mag_field.JPG/… If you look at this diagram which gives a vertical view, then it should make sense – Eliza Jan 19 '14 at 7:11
• Look at i.stack.imgur.com/CsZPJ.png . Isn't the brown arrow contradicting the green arrow? – Pacerier Jan 19 '14 at 7:39
• @Pacerier: the green arrow and brown arrow are the same. The magnetic field is going out of the page and the going in to the page – Eliza Jan 19 '14 at 8:02
• the green arrow shows the field going out of the page from the top, and entering the page from the bottom. However, the brown arrow shows the field going out of the page from the bottom, and entering the page from the top. That's two opposite directions isn't it? – Pacerier Jan 19 '14 at 8:07

The magnetic fields due to the two opposing currents DO in fact cancel to a certain extent - when you get far enough away. Remember (Biot-Savart) that the magnetic field for an infinitesimal wire segment drops off as $1/r^2$, where $r$ is the distance to the wire. Taking a single loop for a second, we can see that at the CENTER of the loop, the field due to each of the elements of the wire add up - they all point down, for example. However, once you are OUTSIDE the loop, they point in opposite directions - the field due to the nearest wire points up, while the one due to the furthest wire points down. However, the distance to the nearest vs the furthest wire element is not exactly the same - the difference is $2r$. The cancellation of the two fields gets better as you get further away - as the distance gets larger than $r$. In fact you end up with the field falling off as the cube of the distance: $1/r^3$. This is typical of a dipole field.

A similar thing happens with electrical charges: when you bring two equal and opposite charges close together, their fields don't exactly cancel, but it does drop off quickly with distance: once again, an inverse cube law.

So your intuition is right - the currents cause cancellation outside of the solenoid; but that cancellation isn't perfect, and this is how the field lines can be "closed".

No it'll not get canceled. Imagine the coil in your mind. Imagine that you are viewing the coil from the top, in your example the current is in clock wise direction. Imagine that you're holding the wire with your right hand with your thumb towards the direction of the current. Now imagine that you are revolving around the coil with your hand still holding the wire you can see that your remaining fingers are along the same direction anywhere around the circle.

A example for this is, assume that there is a train to reach the centre of the earth with the spiral track. Here you are sitting on the right side window seat of the train. Assume that you have a defect in your neck so that you can never see upwards. Now you looking through the window(obviously downwards). You can observe that you can never never see the sky in your journey to the centre of the earth. If you think you can see the sky then you can say the there is no magnetic field inside the coil because of cancellation. I think this example is big. You can think of a spiral staircase.

• If I revolve around the coil with my hand still holding the wire, my fingers are not along the same direction anywhere around the circle. The direction is "downwards towards me" when the current moves left, and "downwards away from me" when the current moves right. – Pacerier Jan 19 '14 at 4:08
• Ok, revolve around the coil with your forefinger not circling the wire. Remember that there is nothing called left or right in a circle there is only clockwise and anti clockwise. – Akshay Nagraj Jan 19 '14 at 4:13
• The direction of the magnetic field is clockwise as the current in the coil moves towards west, and the direction of the magnetic field is anti-clockwise as the current in the coil moves towards east. Why doesn't the clockwise magnetic field "cancels out" the anti-clockwise magnetic field? – Pacerier Jan 19 '14 at 4:16