GREPhysics.NET: GR9277 #9

GR9277 #9

Problem

GREPhysics.NET Official Solution

\prob{9}

A coaxial cable having radii a, b, and c carries equal and opposite currents of magnitude i on the inner and outer conductors. What is the magnitude of the magnetic induction at point P outside of the cable at a distance r from the axis?

Zero
$\frac{\mu_0 i r}{2\pi a^2}$
$\frac{\mu_0 i}{2\pi r}$
$\frac{\mu_0 i}{2\pi r}\frac{c^2-r^2}{c^2-b^2}$
$\frac{\mu_0 i}{2\pi r}\frac{r^2-b^2}{c^2-b^2}$

Electromagnetism $\Rightarrow$ }Current Directions

The opposite currents cancel each other, and thus the induction (and field) outside is 0.

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Comments

pam d
2011-09-27 22:38:58

After having seen these practice tests a few times, I can tell anyone that reads this comment that coaxial cables are definitely something to know.

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none
2008-10-18 14:02:10

Or more theoretically, the system has axial symmetry, so on a circle centered around the cable, $\vec{B} = constant$ . $\oint_C \vec{B} d\vec{l} = \mu_0 i$ . Enclosed $i$ is zero, so $\vec{B}$ must be zero.

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Poop Loops
2008-10-05 13:07:57

If you don't remember what coaxial cables are for (shielding a current), then you can do it the following way:

The field has to go to Zero at r -> inf (nothing else makes sense), which only leaves choices A (since it's always Zero), and C.

From here you have a good shot at guessing, but if you remember that C is the field for an unshielded wire (Ampere's Law) http://teacher.pas.rochester.edu/phy122/Lecture_Notes/Chapter31/chapter31.html

So A is the only choice left.

ramparts
2009-08-09 16:44:15

The other good way to eliminate C is that it's independent of a and b. The only reason to even suspect this answer is non-zero is because a and b are in different places (and presumably because there's more b surface area than a surface area), so it's ridiculous to have an answer that doesn't involve these two quantities.

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kevglynn
2006-10-14 10:10:11

If you're like me, you've been through an entire undergraduate physics degree without ever hearing the Magnetic Field referred to as the "magnetic induction." I thought they were talking about electromagnetic induction or something else of which I hadn't heard.

Magnetic Induction is sometimes used for Magnetic field so as not to mix up H and B. I always learned that H was the Auxilliary Field, and B, the Magnetic Field, so be careful if you're like me, and don't slip because of mere semantics.

Furious
2007-10-27 19:08:16

Yeah man, I was in the same boat. I was sitting there going like, "Magnetic Induction, what does that have to do with this system." But in the end I remembered that the entire point of coax cables is to keep the E&M inside the wire, so I got this one right.

Jeremy
2007-10-29 13:43:31

I too was ignorant of the term, and yet, at some time, I must have read what Griffiths has on page 271: "Many authors call $\vec{H}$ , not $\vec{B}$ , the 'magnetic field.' Then they have to invent a new word for $\vec{B}$ : the 'flux density,' or magnetic 'induction' (an absurd choice, since that term already has at least two other meanings in electrodynamics)." Oh well.

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If you don't remember what coaxial cables are for (shielding a current), then you can do it the following way: The field has to go to Zero at r -> inf (nothing else makes sense), which only leaves choices A (since it's always Zero), and C. From here you have a good shot at guessing, but if you remember that C is the field for an unshielded wire (Ampere's Law) http://teacher.pas.rochester.edu/phy122/Lecture_Notes/Chapter31/chapter31.html So A is the only choice left.

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