Why is there a difference between the magnitude of magnitude field of a finite wire in contrast to an infinite wire?

Besides the two equations(from Ampère's Law & Biot-Savart Law) why is there a weaker magnetic field produced by a finite wire than the infinite wire? If we experimented between a wire(A) and another wire that is 100x wire(A) at wanted to measure the magnitude of the magnetic field at an equal point(P) the longer wire would produce a greater magnitude.

If the parameters are set to have an equal current, and wire shape(expect length), why would they differ?

Your mistake appears to be to believe that the magnetic field produced by a short piece of current-carrying wire is only non-zero at points lying in a plane perpendicular to the wire, going through the wire segment.

That is not the case. The Biot-Savart law tells us that each current-carrying wire segment contributes a magnetic field, given in vector form by $$d\vec{B}(\vec{r}) = \frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \vec{r'}}{|\vec{r'}|^3},$$ where $\vec{r}$ is the position vector in space where you wish to know the B-field and $\vec{r'}$ is a vector from a point on the wire to that position in space. That is, the magnetic field produced by each wire segment is in a direction given by the vector product of the wire segment's instantaneous direction and a vector between the wire segment and the point in space where you wish to calculate the B-field. Even a long way along the infinite wire from where you have marked point $P$, there will be a contribution, because $d\vec{l} \times \vec{r'}$ only approaches zero asymptotically with distance from $P$.

The total magnetic field then needs to be calculated by summing up (integrating) over the contributions from all wire segments in a vector fashion. Since the short piece of wire contains wire segments that are just a subset of those making up an infinite piece of wire, then the B-field from the short piece of wire must be smaller.

Each element of a current carrying wire contributes to the magnetic field at a position in space.

So all the elements of a finite wire combine to produce a certain value for the magnetic field at a point in space.
All the elements of an infinite wire which are outside the length of the finite wire will contribute and produce a magnetic field which is larger than that for a finite wire.

This is really Biot-Savart with the limits of integration being dictated by the ends of a finite wire compared with the limits being plus/minus infinity.
More elements contribute to the magnetic field at a given position for an infinite wire.

As with all such experiments for a wire of finite length one has the problem of the additional magnetic fields produced by the wires connected to the wire under test.

• I think where I struggle to grasp your point is when I imagine how all the current elements would contribute to the magnetic field? If you'd look at @Farcher the added diagram in the question, I only see the cross sections of each relative to that point, if the field produced by each would be perpendicular to length, how would the far left of the inifite wire contribute to the center point P? Rather than the cross section of the wire parallel to it? Jan 1, 2017 at 14:47
• @XCIX I don't understand why you mention cross-sections, nor what cross-sections you are referring to, Jan 1, 2017 at 19:55