# Is Ampere's Law Affected by External Currents?

• ubergewehr273
In summary, The Ampere's circuital law states that when trying to find the magnetic field due to a set of current carrying wires, an imaginary Amperian loop is drawn and Ampere's law is used to find the magnitude of the magnetic field. The RHS involves only the enclosed current inside the loop, but the net magnetic field is due to both the current carrying wires inside and outside the loop. This is similar to Gauss's law for a uniform spherical charge distribution. In a similar scenario, when using Ampere's law with an Amperian loop outside of the current distribution, the B-field is only due to the current enclosed within the loop. However, Ampere's law can always be used as one of Maxwell's
ubergewehr273
When we try to find magnetic field due to a set of current carrying wires in a region we draw an imaginary amperian loop and using ampere's law find the magnitude of the magnetic field.
##\oint \vec B \cdot d\vec l = \mu_{0}i_{enclosed}##
The RHS involves only the enclosed current inside the loop but however the net magnetic field is due to the current carrying wires both inside and outside the loop. I don't seem to understand this part. Because in another scenario if I have the same current carrying wires to be enclosed in the loop that were enclosed in the previous case but now I remove all the wires that are present outside the loop. Now the net magnetic field must change but when I apply ampere's law, I still get the same magnitude of magnetic field as the previous case. Isn't this contradicting the law ?

Ashes Panigrahi said:
When we try to find magnetic field due to a set of current carrying wires in a region we draw an imaginary amperian loop and using ampere's law find the magnitude of the magnetic field.
##\oint \vec B \cdot d\vec l = \mu_{0}i_{enclosed}##
The RHS involves only the enclosed current inside the loop but however the net magnetic field is due to the current carrying wires both inside and outside the loop. I don't seem to understand this part. Because in another scenario if I have the same current carrying wires to be enclosed in the loop that were enclosed in the previous case but now I remove all the wires that are present outside the loop. Now the net magnetic field must change but when I apply ampere's law, I still get the same magnitude of magnetic field as the previous case. Isn't this contradicting the law ?

But you HAVE seen something similar to this before, haven't you? Look at Gauss's law for a uniform spherical charge distribution, for instance. The E-field at points on the surface is due ONLY to charge enclosed INSIDE the gaussian surface, and not due to the charges outside of the surface. This is the E-field ONLY at points on the surface, not everywhere in space. You will have to do another gaussian surface to represent regions outside of the charge distribution.

The Ampere's circuital law is similar to such a concept. The B-field around the loop represented by the Ampere's loop is only due to the current enclosed within the loop. But as with Gauss's law, this is the B-field only on that loop. If you want to find the B-field outside of the current distribution, then you have to do another Amperian loop OUTSIDE of the current volume.

Zz.

Ashes Panigrahi said:
The RHS involves only the enclosed current inside the loop but however the net magnetic field is due to the current carrying wires both inside and outside the loop.
It is important to note that the LHS is not just the magnetic field, but is the integral of B.dl. So, let's consider a uniform "external" field, such as a solenoid located outside of the loop. Because of the form of the integral the result is 0, regardless of the value of the uniform field and therefore regardless of the current in the solenoid outside of the loop. Although it is not obvious, the same thing happens for a non-uniform field also.

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Dale said:
It is important to note that the LHS is not just the magnetic field, but is the integral of B.dl.
The whole point of Ampere's law is to use the power of symmetry to reduce the problem to simple cases so as to make it easy to find the magnitude of the B field. But then what's the point of finding integral of B.dl if the B field itself is different in cases where external magnetic field is present outside the loop and otherwise.

Ashes Panigrahi said:
The whole point of Ampere's law is to use the power of symmetry to reduce the problem to simple cases so as to make it easy to find the magnitude of the B field. But then what's the point of finding integral of B.dl if the B field itself is different in cases where external magnetic field is present outside the loop and otherwise.
Well, it only works for that purpose in highly symmetric situations, such as close to long straight wires or inside a coaxial cable. It cannot be used for that purpose in general cases.

However, it can always be used as one of Maxwell's equations, and the differential form is more generally useful as well.

Dale said:
Well, it only works for that purpose in highly symmetric situations, such as close to long straight wires or inside a coaxial cable. It cannot be used for that purpose in general cases.

However, it can always be used as one of Maxwell's equations, and the differential form is more generally useful as well.

Actually, just like Gauss's Law, it can be used in any general situations. It is just that it can't be solved easily or analytically for non-symmetric configurations. There's nothing to stop anyone from solving it numerically for non-symmetric situations.

Zz.

Delta2 and Dale
ZapperZ said:
Actually, just like Gauss's Law, it can be used in any general situations. It is just that it can't be solved easily or analytically for non-symmetric configurations. There's nothing to stop anyone from solving it numerically for non-symmetric situations.
Yeah, good point. You are right.

I still haven't got my doubt clarified. How does the B field change whether I place an external current carrying conductor or not ?

Ashes Panigrahi said:
I still haven't got my doubt clarified. How does the B field change whether I place an external current carrying conductor or not ?
Hmm, I thought I already explained above:

Dale said:
the LHS is not just the magnetic field, but is the integral of B.dl. So, let's consider a uniform "external" field, such as a solenoid located outside of the loop. Because of the form of the integral the result is 0, regardless of the value of the uniform field and therefore regardless of the current in the solenoid outside of the loop.

Dale said:
I didn't quite get your point.
Dale said:
It is important to note that the LHS is not just the magnetic field, but is the integral of B.dl. So, let's consider a uniform "external" field, such as a solenoid located outside of the loop. Because of the form of the integral the result is 0, regardless of the value of the uniform field and therefore regardless of the current in the solenoid outside of the loop. Although it is not obvious, the same thing happens for a non-uniform field also.
Then what is the point of finding integral B.dl if effects of external magentic field gets ignored?

Ashes Panigrahi said:
Then what is the point of finding integral B.dl if effects of external magentic field gets ignored?
It tells you what is the effect of the sources on the field. By itself it does not completely and uniquely tell everything about the magnetic field, but it does tell something important.

Dale said:
It tells you what is the effect of the sources on the field. By itself it does not completely and uniquely tell everything about the magnetic field, but it does tell something important.
What do you mean by "sources"?

Ashes Panigrahi said:
What do you mean by "sources"?
The currents are the sources. Ampere’s law describes how currents are related to the magnetic field.

Dale said:
The currents are the sources. Ampere’s law describes how currents are related to the magnetic field.
Ok thanks a lot.

Dale

## 1. What is Ampere's law?

Ampere's law states that the magnetic field around a closed loop is directly proportional to the electric current flowing through the loop.

## 2. Why is there doubt about Ampere's law?

There is doubt about Ampere's law because it does not always accurately predict the magnetic field around a current-carrying wire. This is known as the "Ampere's law paradox."

## 3. How can Ampere's law be explained?

Ampere's law can be explained by taking into account the displacement current, which is a time-varying electric field that can create a magnetic field.

## 4. Is Ampere's law still used in scientific research?

Yes, Ampere's law is still used in scientific research, but it is often modified to account for the displacement current and other factors that were not initially considered.

## 5. Are there any alternative theories to Ampere's law?

Yes, there are alternative theories such as the Biot-Savart law and the Maxwell's equations, which take into account the displacement current and have shown to be more accurate in predicting magnetic fields.

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