# Application of Biot-Savart's law on a current-carrying wire of finite length

In this figure, how will the wire elements between O&a and O&-a affect the magnetic field at point 'P' which lies at a distance x from point O? Because none of the field lines created due to the wire elements between points a and O and points -a and O seem to pass through point P. Maybe I'm lacking some important concept. Consider a point 'P' at some horizontal distance 'x' from the middle point of the wire. If we draw the magnetic field lines due to an infinitesimally small current-carrying wire element, then they don't seem to pass through point 'P' but still, we say that it affects the magnitude of the magnetic field vector at point 'P'...why?...because in order to do so the field lines due to all the tiny elements on the wire should pass through point 'P'...

• They definitely do pass through point P, also biot savart isn't valid since the wire is finite Jul 12, 2022 at 5:59
• . . . also biot savart isn't valid since the wire is finite . . . is news to me as when using BS one starts off with a line segment, ie an element of finite length. Is it a confusion with Ampere's law? Jul 12, 2022 at 7:22
• Biot and Savart is valid in this case. The field generated by a segment like this one is a very classic example that is shown in most textbooks about magnetostatics. This example is, however, quite artificial since the current has to go somewhere, but that's another problem. Jul 12, 2022 at 8:35
• What is the divergence of this current density function? I am almost 100% sure that $\nabla \cdot \vec{J} ≠ 0$. As such, biot savart law for this case does not satisfy amperes law. Unless you claim that even in this situation it satisfies the ampere maxwell law? Jul 12, 2022 at 9:10
• When $\nabla \cdot \vec{J} ≠ 0$, biot savart does not satisfy amperes law, this can be shown by taking the curl of biot savart. And is a direct consequence of taking the divergence of amperes law aswell. Jul 12, 2022 at 9:11

The field lines are concentric circles in all plains perpendicular to the current density shaded in purple.

I have drawn smaller circles to indicate magnitude, but they extend to infinity.

Hence the field from the current density reaches point p

• Thank you for the diagram!...It would be great if you could help me visualize the field lines passing through point 'P' in the 2nd diagram I've attached. Jul 12, 2022 at 16:35
• I'm not sure if I can be more clear than the diagram I have already drawn the magnetic field vector coming from a specific wire element, at point P, is still going to make concentric circles, with the direction following the standard right hand rule, it's just the magnitude is going to be different. Jul 12, 2022 at 18:45
• physics.stackexchange.com/questions/545027/… here is my derivation of the magnetic field of a point charge, assuming amperes law holds (which is does NOT, neither does the situation you've described, you cannot use biot savart) none the less, it is still usefull in determining the field of a single element aka a moving individual charge. Here the B field is in the direction of the cross product, it is perpendicular to the current direction, and the r unit vector coming radially from the point charge. Jul 12, 2022 at 18:47
• Imagine limiting r to one plane. Drawing a circle around the charge, given the current points into the screen. By definition that magnetic field is perpendicular to the r unit vector, so must make a circle given we evaluate the field on a circle around the charge. This direction is still preserved if we extend the evaluation of the field, but into the page (a cylinder shape), the magnitude just decreases as the 2 vectors get more and more parrallel. The magnitude also decreases as 1/r^2 as you move further away from the current element. Jul 12, 2022 at 18:50
• Point 'P' is a single point. Jul 13, 2022 at 7:26

Physically speaking the current could not just end like you show. There should be charging on the apex of the wiring, otherwise wire should go somewhere. Anyway the Biot-Savare law stands that in vacuum

$$$$\vec{B} = \frac{\mu_0} {4\pi}\int_{wire} \frac{I \vec{dl}\times \vec{r'} }{ |\vec{r'}|^2 }$$$$ where $$\vec{r'}$$ is the vector from the local current to the observation point. So if you have a finite current source you could formally integrate over your finite wire in order to get the magnetic field.

• Thank you for your response!! but my main confusion is why any other arbitrary point on the current-carrying wire should affect the magnitude of the Magnetic field at the observation point because if choose any arbitrary point on the wire(Other than point 'O') and draw the magnetic field lines (which are in this case concentric circles), we won't see them pass through the observation point. Jul 12, 2022 at 8:36
• @Dexter, I am not sure to understand. Do you have a confusion on the magnetic field vector orientation, is that it? Jul 12, 2022 at 8:45
• No, I'm not having any confusion related to the Magnetic Field vector orientation. Rather my confusion is that when we consider a current-carrying wire, we say that every infinitesimally small current-carrying element on the wire of length, say dl will affect the magnitude of the magnetic field vector(At the point of observation)...My question is why...Because if we draw the Magnetic field lines from every point on the wire (Except point O), we won't see our point of observation lie on any one of them(Field lines). Jul 12, 2022 at 8:56
• Magnetic field lines from a single element do not just radiate perpendicular to the current elements vector, it radiates in a sphere. Why do you think otherwise? Jul 12, 2022 at 9:03
• There is still magnetic field generated by the wire, that is formally not 0 at point P following the equation. (and consequently there are magnetic field lines). The magnetic field lines from the elementary current will be as drawn by @jensen paull. The total magnetic field is the superposition of the elementary magnetic fields, and each wire has a different contribution. Jul 13, 2022 at 7:39