Magnetic field generated by a wire crossed by current: clarifications

Question

We know that from Oersted's experience the lines of force of the magnetic field generated by a wire crossed by current $I$, are concentric circumferences of radius $r$ (variable) where the centre is a point of the wire.

To prove that these are actually circumferences I have followed this path. The law of the magnetic field of a current crosses the wire is:

$$B(r)=2k_m\frac{I}{r}, \quad \tag{1}$$

where $k_m=k_e/c^2$. Now from $(1)$ we have also:

$$r=2k_m\frac{I}{B(r)}, \quad \tag{2}$$

If we indicate with

$$K=2k_m\frac{I}{B(r)}, \quad \tag{3}$$ we have:

$$r=K, \quad \tag{4}$$

But if $r=\sqrt{x^2+y^2}\,$ ($2-$dimensions plane) then from $(4)$, $$x^2+y^2=K^2$$ which is exactly a circumference with center in in the origin (a point of the thread where we have fixed a Cartesian orthogonal reference system).

If $r=\sqrt{x^2+y^2+z^2}\,$ ($3-$dimensions space) should I draw spheres with a thread point in the middle or are they always circumferences with the center of a point of the wire?

Answer 1

You said:

If $ r = \sqrt{x^2+y^2+z^2} $ (3−dimensions space) should I draw spheres with a thread point in the middle or are they always circumferences with the center of a point of the wire?

But it simply isn't, in that formula $r$ is defined as the distance between the point in which you evaluate the field and the wire, in the same manner as you define the distance between a point and a straight line.

I really don't understand what you are doing in your calculation.

In the first place, I want to clarify a point, what I think you are trying to find with your calculation is the set of points in which the field has the same intensity, but this is not the definition of lines of force, maybe you are making some confusion.

Moreover, even if you want to find the set of points in which the field has the same intensity, let me redo that in a cleaner way.

You have to start from the assumption of costant intensity of the field:

$$ B(r) = C $$

where C is a costant. Then, for the law you mentioned:

$$ 2k\frac{I}{r} = C $$

So:

$$ r = 2k\frac{I}{C} $$

If one further assumes that I is costant, then r is costant too:

$$ r = C' $$

Being $r$ defined as $\sqrt{x^2 +y^2}$ then you have:

$$ \sqrt{x^2 + y^2} = C' $$

That is the equation of a circumpherence.

So, this is a correct process to get the set of points in which the field has the same intensity, but I stress again that this are not the lines of force.

The lines of force associated to a vector field are defined (in an elementary way) as those lines such that the vector field is always tangent to them, in this case this two distinct concepts coincide, but it is a mere coincidence.