Question

Equation 1 is derived for thin wires - wires whose diameters are very small compared to their distance apart. However, it is not very accurate for wires whose diameters are comparable to their separation distance (as in this experiment). Why? Use a sketch to explain. Equation 1 is∮B.dL=μoI

Answer 1

the "integral form" of the original Ampère's circuital law is a line integral of the magnetic field around some closed curve C (arbitrary but must be closed). The curve C in turn bounds both a surface S which the electric current passes through (again arbitrary but not closed—since no three-dimensional volume is enclosed by S), and encloses the current. The mathematical statement of the law is a relation between the total amount of magnetic field around some path (line integral) due to the current which passes through that enclosed path (surface integral). It can be written in a number of forms.

In terms of total current, which includes both free and bound current, the line integral of the magnetic B-field (in tesla, T) around closed curve C is proportional to the total current I_enc passing through a surface S (enclosed by C):

$\oint_C \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 \iint_S \mathbf{J} \cdot \mathrm{d}\mathbf{S} = \mu_0I_\mathrm{enc}$

So in mathematical terms, the diameter must be small compared to the length so that the line and surface integrals could be applied (to use a line integral one must use an actual line path, if the wire has a certain diameter this no longer applies)

Also with the force law:

The best-known and simplest example of Ampère's force law, which underlies the definition of the ampere, the SI unit of current, states that the force per unit length between two straight parallel conductors is

$\frac {F_m} {L} = 2 k_A \frac {I_1 I_2 } {r}$,

where k_A is the magnetic force constant from the Biot–Savart law, F_m/L is the total force on either wire per unit length of the shorter (the longer is approximated as infinitely long relative to the shorter), r is the distance between the two wires, and I₁, I₂ are the direct currents carried by the wires.

This is a good approximation if one wire is sufficiently longer than the other that it can be approximated as infinitely long, and if the distance between the wires is small compared to their lengths (so that the one infinite-wire approximation holds), but large compared to their diameters (so that they may also be approximated as infinitely thin lines).