Edited.TSny said:
I though length element should be parallel to current?TSny said:
Review the mathematical statement of Ampere's law. Be sure that you can state the meaning of all of the symbols that occur in the mathematical statement of the law. That should help you see why the vector ##\vec{ds}## does not have to be in the direction of the current.Nway said:
Sir my textbook says that vector ##\vec{ds}## is just a small length element. Nothing to do with direction of current?TSny said:
Please give your textbook's statement of Ampere's Law .Nway said:
Ok I'll find it.SammyS said:
SammyS said:
Because the plane of the circle in which this element lies is perpendicular to the current. Any vector lying in that plane has, by definition, no component parallel to the current.Nway said:
Sorry sir. What do you mean the plane of current? I don't understand because when there is a current in a wire like here:kuruman said:
Look at what Ampere's law says. You are not integrating along the current carrying wire. The wire passes through an open surface. The line integral is along a (closed) path which is along the boundary of the open surface.Nway said:
Thank you sir. I see now maybe.nasu said:
Sir thank you. I assume that ##d\vec{s}## has same meaning for Ampere's law as with Biot–Savart law? I am now correct to say that ##d\vec{s}## gives direction of path at each differential point along a path.SammyS said:
That's a different ##d\vec s## from the one you asked about in the original post. Look at your picture below that you yourself circled in yellow. It shows a circle defined by a blue dotted line and current ##I## with an arrow in the direction of the current. The current is perpendicular to the circle and to all vectors in the plane of that circle including the ##d\vec s## shown therein.Nway said:
Thanks sir. So why dose ##d\vec{s}## have to be perpendicular to the current (in the plane of the circle)? Maybe a better restatement of my confusion: Is there a generalized definition for ##d\vec{s}## that is is Bio-Savart and Ampere's law?kuruman said:
Thank you very much sir. I still don't understand why ##d\vec{s}## is tangent to path. It is similar to how the velocity vector is tangent to a circular path in uniform circular motion?nasu said:
Yes, it is similar to that.Nway said:
Sir thank you. What are the differences?kuruman said:
The velocity is tangent to the path for any motion. Do you know the definition of the velocity vector?Nway said:
Yes I do sir. Velocity vector is time integral of position vector.nasu said:
Yep sorry sir I got it the wrong way round.nasu said:
Yes, it is true that Ampere's law does not demand anything in the way of symmetry.nasu said:
Can ##\vec{ds}## also be though of as the displacement of a positive test using the right hand rule?nasu said:
I don't know what you mean by this.Nway said:
No, ##\displaystyle \vec{ds}## cannot be thought of as the displacement of a positive test charge, using the right hand rule?Nway said:
The length element in Ampere's law, often denoted as \( \mathbf{dl} \), is a differential vector element along a chosen path or loop in a magnetic field. It represents an infinitesimally small segment of the path, used in the line integral to calculate the circulation of the magnetic field around the loop.
In the integral form of Ampere's law, the length element \( \mathbf{dl} \) is used to perform a line integral of the magnetic field \( \mathbf{B} \) around a closed loop. The law is expressed as \( \oint \mathbf{B} \cdot \mathbf{dl} = \mu_0 I_{\text{enc}} \), where \( \mu_0 \) is the permeability of free space and \( I_{\text{enc}} \) is the current enclosed by the loop.
The path for the length element in Ampere's law is chosen based on symmetry and convenience to simplify calculations. Common paths include circles, rectangles, or other shapes that match the symmetry of the magnetic field and current distribution. The chosen path should ideally make the integral easier to evaluate.
The direction of the length element \( \mathbf{dl} \) in Ampere's law is tangent to the chosen path at each point and follows a specific orientation, usually defined by the right-hand rule relative to the current direction. This orientation is crucial for correctly applying the dot product in the line integral.
In the differential form of Ampere's law, the length element \( \mathbf{dl} \) is not explicitly used. Instead, the law is expressed using the curl of the magnetic field: \( \nabla \times \mathbf{B} = \mu_0 \mathbf{J} \), where \( \mathbf{J} \) is the current density. The differential form relates the local properties of the magnetic field directly to the current density.