Why does Ampere's law give the same result for finite and infinite conductors?

[Dr.Brain]

If we want to find out the net magnetic field on a point near a conducting wire , we use ampere's law by take a closed surface in form of A CIRCLE...with radius equal to the distance of the point fromt he where...but where ampere fails is that it gives the same result for both a finite conductor and infinite conductor because the formulae does not deal with the length of the conductor.

 

[Hurkyl]

You'd have a point if you could show the length of the conductor mattered.

 

[maverick280857]

Ampere's Law states that

$$\oint B dl = \mu_{o}I_{enc}$$

If you look at the expression for Ampere's Law (as it is written above) and compare it with Gauss's Theorem in Electrostatics which states that

$$\oint E dS = \frac{Q_{enc}}{\epsilon}$$

you will find the degree of closeness in these expressions. This closeness is explained mathematically using very important theorems in Vector Calculus (Green's Theorem or Gauss's Theorem). These theorems hold for vector functions in general and there is nothing very special about electric field or magnetic fields except one extremely special property of electrostatic electric fields, which is that the line integral of E taken over a closed path is zero, i.e.

$$\oint E dl = 0$$

If you understand these basic facts, you should have no trouble in understanding why the Gauss's Theorem is "inaccurate" from the point of view that you have. Suppose I want to determine the electric field due to an infinite line of charge. Let's say I do this using Gauss's Law. Utilizing the fact that the electric field is radial and uniform, I choose a Gaussian surface which is a cylinder of radius r and length L concentric with the wire. If $$\lambda$$ be the linear charge density associated with the wire, then $$Q_{enc} = \lambda L$$ is the charge enclosed by it. The integral,

$$\oint E dS = \frac{Q_{enc}}{\epsilon}$$

simplifies to

$$E \oint dS = \frac{\lambda L }{\epsilon}$$

The surface integral is simply the surface area of the cylinder S where

$$2\pi r LE = \frac{\lambda L}{\epsilon}$$

At this stage you should see a funny thing: I choose L to be the length of my Gaussian surface, but L cancels out and the expression I get for E after that is what I call E for the entire wire! How can I say THAT? When I didn't integrate over the length of the wire! Most of all, WHY DID I CANCEL L? L is supposed to be infinite isn't it! How can I cancel a quantity that is infinite, on both sides of an equation?

Thats what you should be worrying about. Now try and use this idea in Ampere's Law to find the magnetic field outside an infinite conducting wire carrying a current I. You will reach a similar conclusion.

If you're still wondering why this "paradox" has crept in into Physics, you should know that it isn't a paradox if I write dL instead of L, because mathematically I am allowed to subdivide an infinite length into infinitesimal elements of size dL. After all dL is neither zero nor infinity! It tends to zero but it isn't quite zero you see. Calculus is correct here. I should write EdS = (1/epsilon)*dQ and write dQ = lambda * dL and dS = 2*pi*r*dL. That way, dL cancels out and I get a correct expression.

Alternatively I could define L as a very small quantity that is not zero (nor infinite). This seems a hardly convincing explanation but cancelling out L is the only reason which makes you think Ampere's Law--or for that matter Gauss's Law--does not involve the length of the conductor. It doesn't :-)...

Cheers
Vivek

 

[Arctic Fox]

Man... I really need to catch up on math.

All these cool mysteries go right over my head.

;)

 

[Chronos]

You'd have a point if you could show the length of the conductor mattered.

Duh? That somehow made the case that length is matters?

 

[Locrian]

Who moved this? He's absolutely right and wrong at the same time, and this theory is NOT under deveolopment. Maxwell solved it already.

Imagine a short wire with a current running through it. Would the magnetic field be the same as if it were infinite? The answer is no. As charges traveled down the finite wire, they would build up at one end. Instead of having a steady electric field within the imaginary circle, the field would change.

Ampere's Law originally stated that the magnetic field around the wire was equal to uI. However, maxwell added a term that affects the magnetic field if the electric field is changing.

Therefore, the original poster is correct, Ampere's law did originally come up short (he was probably taught just the one part in school). However, Ampere's Law as it is known now is fully functional. If he knew Maxwell's equations he would have known there is no longer any paradox.

 

[maverick280857]

So Maxwell's Equations will provide a correct explanation for the "inaccuracy" (which doesn't exist as Locrian has pointed out). However, in school, we were taught Ampere's Law the wrong way (similarly, displacement currents were hard to come to terms with because the equation of continuity was never taught either!). This is probably why Dr Brain thinks Ampere's Law is faulty...it is indeed faulty in the "B dot dl = mu_not n I" form.

By the way Dr. Brain for more info on this, you might want to refer to the chapter on Magnetic Properties of Materials in Resnick and Halliday's classic physics textbook volume #2. The extra term has been spoken of in this chapter.

 

[Dr.Brain]

j thnx maverick...i just found that ampere's law is really good in solving problems in which there is a cavity in a a conductor...

 

[maverick280857]

Cavity in a conductor? As far as I can remember, I've encountered such problems in electrostatics. Are you doing magnetism from an EM theory book? If you are preparing for JEE, I do not think the mathematical complexity will be so large...yet, its always a good thing to know something from first principles.

Cheers
Vivek