Magnetic field inside hole in half a wire

In summary, the conversation discusses the use of Ampere's law to calculate the magnetic field inside an infinitely long wire with a hole cut out of it. It is concluded that the integral path must agree with the direction of the magnetic field for Ampere's law to hold, and that a closed loop integral of the magnetic field does not necessarily imply a zero magnetic field.
  • #1
Nick89
555
0
This is not really a homework problem, just something that struck me as weird while revising for an exam...

Homework Statement


Consider an infinitely long straigt wire with an also infinitely long hole cut out of it.
Then imagine cutting the wire in half, as in the following picture:
10i7b0m.jpg


The 'wire' carries a current I along it's axis.

Now, according to Ampere's law for magnetic fields (see below) there should be no magnetic field inside the hole, right?

Why is this?

I can imagine it if the wire was not cut in half (just a wire with a hole in it) since than the field would cancel itself out (top side cancels out lower side, left side cancels out right side, and so on for every 'pair' of sides).

Now however, that is not the case... So how come there is still no magnetic field in the hole? I just can't imagine this, I think it's weird...

Am I not seeing something obvious maybe?

Homework Equations


Ampere's law:
[tex]\oint \vec{B} \cdot \vec{dl} = \mu_0 I_{\text{enclosed}}[/tex]

The Attempt at a Solution


Using Ampere's law as stated above, with the integral path shown in red in the picture, I find the enclosed current to be 0, resulting in a magnetic field B of also 0...

Is this right, or am I doing something horribly wrong?!
 
Physics news on Phys.org
  • #2
Alright I think I figured it out... It would really help me if someone could verify this for me though please!

In the case of a 'normal' wire (without any hole and not cut in half) the magnetic field would be in a circle around the wire's axis. That's why you use a circular integral path when using Ampere's law. In other words, the path you chose has to 'agree' with the direction of the field you're trying to calculate.

Now however, the magnetic field is probably no longer circeling around the axis (at least not uniformly) so the integral path does not agree with the direction of the magnetic field and I cannot use this method.

Am I correct in saying this? Should I use Biot-Savart's law for example instead?(Note, I'm not trying to actually calculate the field, not for now at least. All I'm trying to do is understand why Ampere's law gives me a 0 magnetic field while my intuition says that can't be true.)
 
  • #3
for ANY wire, if you integrate around a path that does not go round the wire, you will get 0. That doesn't mean there is no magnetic field.
 
  • #4
Hm I guess your right about that, quite obvious now I think about it.
When you take the circular path at a distance from the wire you also get 0 which is obviously wrong...

But, does that mean my last statement is correct? Must the path 'agree' with the direction of B for ampere's law to hold?

In other words, what are the conditions the path must fullfil for ampere's law to be valid?
 
Last edited:
  • #5
Ampere's law is always valid. If the path doesn't go round the wire then [tex]
\oint \vec{B} \cdot \vec{dl} = \mu_0 I_{\text{enclosed}} = 0
[/tex]
 
  • #6
Yeah I understand that.

I think I got it now. If [tex]\oint \vec{B} \cdot \vec{dl} = 0[/tex] that does not imply that B is 0, right? It could also be that the closed loop integral dl is 0 (which in this case it is since the point you are calculating from (the wire) is not inside the path).

Correct?
 
  • #7
Nick89 said:
Yeah I understand that.

I think I got it now. If [tex]\oint \vec{B} \cdot \vec{dl} = 0[/tex] that does not imply that B is 0, right? It could also be that the closed loop integral dl is 0 (which in this case it is since the point you are calculating from (the wire) is not inside the path).

Correct?
yes indeed.
 

1. What is a magnetic field?

A magnetic field is a region in space where a magnet or a moving electric charge experiences a force. It is created by the movement of electrical charges and is characterized by its direction and strength.

2. How is a magnetic field created inside a hole in a half wire?

A magnetic field inside a hole in a half wire is created by the flow of electrical current through the wire. The current creates a circular magnetic field around the wire, and the hole acts as a concentrator, making the field stronger inside it.

3. How is the direction of the magnetic field inside the hole determined?

The direction of the magnetic field inside the hole is determined by the direction of the current flow in the wire. The right-hand rule can be used to determine the direction of the magnetic field by pointing the thumb of your right hand in the direction of the current flow and curling your fingers, which will give the direction of the magnetic field.

4. What factors affect the strength of the magnetic field inside the hole?

The strength of the magnetic field inside the hole is affected by the magnitude of the current flowing through the wire and the size of the hole. A larger current or a smaller hole will result in a stronger magnetic field.

5. How is the magnetic field inside the hole measured?

The magnetic field inside the hole can be measured using a magnetic field sensor, such as a Hall effect sensor, placed inside the hole. This sensor can detect the strength and direction of the magnetic field and provide quantitative measurements.

Similar threads

  • Introductory Physics Homework Help
Replies
6
Views
282
  • Introductory Physics Homework Help
Replies
3
Views
276
  • Introductory Physics Homework Help
Replies
2
Views
212
  • Introductory Physics Homework Help
Replies
7
Views
1K
  • Introductory Physics Homework Help
Replies
10
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
136
  • Introductory Physics Homework Help
Replies
5
Views
861
  • Introductory Physics Homework Help
Replies
14
Views
943
  • Introductory Physics Homework Help
Replies
11
Views
2K
  • Introductory Physics Homework Help
Replies
3
Views
948
Back
Top