- #1
QuasiParticle
- 74
- 1
Suppose we have an elliptical loop of wire in the x-y plane with constant cross-section. And let's also assume that the cross-section is very small, so we have a thin wire.
$$\vec{r}=\begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} a\cos \theta \\ b \sin \theta \end{bmatrix}$$
A tangent unit vector to the loop is
$$\hat{v}= \frac{d\vec{r}/d\theta}{|d\vec{r}/d\theta|}=\frac{1}{\sqrt{(a/b)^2 y^2 + (b/a)^2 x^2}} \begin{bmatrix} -(a/b) y \\ \phantom{-}(b/a) x \end{bmatrix}$$
When current flows in the loop, it sounds at first like a reasonable assumption that it is of the form
$$\vec{J} = J_0 \hat{v},$$
that is, constant and directed along the tangent. According to the continuity equation for a system with time-independent charge density, the divergence of the current density should vanish everywhere.
$$\nabla \cdot \vec{J} = 0$$
If we compute the divergence of the above current density, it is of the form
$$\nabla \cdot \vec{J} = \frac{ab(b^2-a^2)xy}{(\textrm{something} > 0)}$$
This is identically zero if and only if ##a=b##, i.e. if the current loop is circular. So it seems that we cannot have an elliptical loop with a constant current density. Instead the "constant" part of the current density should have an angular dependence ##J_0 = J_0(\theta)##. I was somewhat surprised to find this out after having (naively) made the assumption ##|\vec{J}|=\textrm{const.}## and I still don't quite understand why the constant current density is not possible. It must have to do with the varying curvature of the loop, so that the current cannot flow perpendicular to the cross-section at all points? I suppose thinking in terms of current density can be misleading. Does anyone have any comments or insight into this issue?
$$\vec{r}=\begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} a\cos \theta \\ b \sin \theta \end{bmatrix}$$
A tangent unit vector to the loop is
$$\hat{v}= \frac{d\vec{r}/d\theta}{|d\vec{r}/d\theta|}=\frac{1}{\sqrt{(a/b)^2 y^2 + (b/a)^2 x^2}} \begin{bmatrix} -(a/b) y \\ \phantom{-}(b/a) x \end{bmatrix}$$
When current flows in the loop, it sounds at first like a reasonable assumption that it is of the form
$$\vec{J} = J_0 \hat{v},$$
that is, constant and directed along the tangent. According to the continuity equation for a system with time-independent charge density, the divergence of the current density should vanish everywhere.
$$\nabla \cdot \vec{J} = 0$$
If we compute the divergence of the above current density, it is of the form
$$\nabla \cdot \vec{J} = \frac{ab(b^2-a^2)xy}{(\textrm{something} > 0)}$$
This is identically zero if and only if ##a=b##, i.e. if the current loop is circular. So it seems that we cannot have an elliptical loop with a constant current density. Instead the "constant" part of the current density should have an angular dependence ##J_0 = J_0(\theta)##. I was somewhat surprised to find this out after having (naively) made the assumption ##|\vec{J}|=\textrm{const.}## and I still don't quite understand why the constant current density is not possible. It must have to do with the varying curvature of the loop, so that the current cannot flow perpendicular to the cross-section at all points? I suppose thinking in terms of current density can be misleading. Does anyone have any comments or insight into this issue?