# Magnetic Field

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Today, in a test, I got the following question.

State the factors on which the magnetic field due to a current carrying straight conductor depends.

My answer to it is that magnetic field is directly proportional to both current and length of the conductor.

But, my classmates answered- Magnetic field would be directly proportional to current and inversely proportional to the distance between the point and conductor.

Now, the reference of a point in second answer seems incorrect because the question asked the factors affecting the magnetic field of a straight current carrying conductor not the factors affecting the magnetic field at a point? Am I right. Wouldn't a reference to a point change the answer.

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I think it would actually be a combination of what both of you wrote because of the Biot-Savart Law:

$\vec{B}=\int\frac{\mu_{0}}{4\pi}\frac{Id\vec{I}\times\hat{r}}{\vec{r}^{2}}$

So you see that it does go as your classmate said. I is on top, and r is on bottom. However the r is squared, so I would probably reword what your classmate said as ... inversely proportional to the square of the distance...

Notice that the Biot-Savart law is not just for a single point, but is general of all points, it is the equation of the field itself.

So you could also reword what he wrote by saying that the magnetic field itself is proportional to current and inversely proportional to the square of the distance, not just the field at a single point.

Now, this is not the whole picture because we are forgetting the integral, which is over all space (or equivalently, just over the relevant region which is the conductor), so that the magnetic field will depend on the shape and size of the conductor. If it's a wire, then it would depend on the length as you said.

So it would really be 3 things. Proportional to current and length, and inversely proportional to the square of the distance.

Edited by spacelike
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I think the strength of the magnetic field does not depend on the length of the conductor. Of course the field extends along the length of the conductor but whether you can say that a longer conductor has "more" magnetism than a shorter conductor I rather doubt. I suppose it rather depends on what is meant by "factors" and how you want to describe the magnetic field.

http://www.pa.msu.edu/courses/1997spring/phy232/lectures/ampereslaw/wire.html

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I think the strength of the magnetic field does not depend on the length of the conductor. Of course the field extends along the length of the conductor but whether you can say that a longer conductor has "more" magnetism than a shorter conductor I rather doubt. I suppose it rather depends on what is meant by "factors" and how you want to describe the magnetic field.

http://www.pa.msu.ed...eslaw/wire.html

But the Biot-Savart law has length included.

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That length in Biot-savart law is for the geometry of the conductor so I think it does not depend on length in this case. Your question did say the geometry was a straight conductor. My answer to the question would be current alone.

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That length in Biot-savart law is for the geometry of the conductor so I think it does not depend on length in this case. Your question did say the geometry was a straight conductor. My answer to the question would be current alone.

I got the test back. My teacher crossed the length factor. When asked, he said that Biot-savart law results in the relation of inverse proportionality to r.

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I'm sorry, I don't understand why the length of the wire doesn't count.

Say we are looking at the case of points along the plane cutting halfway through the length of the wire (so a circle about the origin, where the wire goes through it and extends from -L/2<z<L/2), the axis of the wire is the $\hat{k}$ direction.

Then $\vec{B}=\int\frac{\mu_{0}}{4\pi}\frac{Id\vec{l}\times\hat{r}}{r^{2}}$

$d\vec{l}=dl\hat{k}$

$dl\hat{k}\times\hat{r}=dl \hat{\phi}$

So now we have:

$\vec{B}=\frac{\mu_{0}}{4\pi}\int\frac{I dl}{r^{2}}\hat{\phi}$

$\vec{B}=\frac{\mu_{0}I}{4\pi r^{2}}L\hat{\phi}$

So we can clearly see that the magnetic field will wrap around the wire in the direction of $\hat{\phi}$ (consistent with the right hand rule)

and also that it's magnitude $\frac{\mu_{0}IL}{4\pi r^{2}}$ is directly proportional to the current and length of the wire and inversely proportional to the distance from the wire squared.

(unless of course I made a mistake, then please correct me anyone)

EDIT: I think perhaps in the limit of an infinitely long wire this might reduce to $\frac{\mu_{0}I}{4\pi r}$.. It would make more sense to me if it did, I can't see how this would happen at the moment though. But it does not make sense to me that the magnetic field would not depend on the length of the wire if you had a wire of finite size, so I am fairly confident in the result I obtained above for a finite sized wire.

Perhaps you could argue with your teacher that they didn't specify if the wire was infinite or finite.

Edited by spacelike
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Perhaps you could argue with your teacher that they didn't specify if the wire was infinite or finite.

He said something about that. He said that when calculating magnetic field of a long conductor, we use the relation which has been derived from Biot-Savart law.

He wrote this on my paper- $\frac{\mu_{0}2I}{4\pi r}$

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