- #1
Mbert
- 63
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Dear colleagues,
I have questions regarding Biot-Savart law. From [1], it is shown that the equation (Biot-Savart) is derived from the solution to Poisson's equation (assuming here div A=0)
\begin{equation}
\vec{\nabla}^2 \vec{A} = -\mu \vec{J}
\end{equation}
which is
\begin{equation}
\vec{A}(\vec{r}) =\frac{\mu}{4\pi}\int_V{\frac{\vec{J}{\rm d}^3r'}{\left|\vec{r}-\vec{r'}\right|}}
\end{equation}
where $\vec{r}$ is the position where $\vec{A}$ is evaluated and $\vec{r'}$ is the position where the integral is evaluated.
The first thing that troubles me is the singularity $\left|\vec{r}-\vec{r'}\right|$ when we evaluate the field at the point of integration. For a wire of finite radius, this means that the $\vec{A}$ field inside the conductor is infinite (or am I missing something?). If so, why in books on electromagnetics do we usually replace the conductor by an equivalent filamentary current $I=\vec{J}\cdot{\rm d}\vec{s}$? The field calculated inside the conductor will be different. This can be seen from the solution for the $\vec{B}$ field by using Ampere's equation (for the infinitely-long finite-radius wire)
\begin{equation}
B_{\theta}=\frac{\mu I}{2 \pi \rho}
\end{equation}outside the wire
\begin{equation}
B_{\theta}=\frac{\mu I \rho}{2 \pi R_{wire}^2}
\end{equation} inside the wire
where $R_{wire}$ is the cross-section radius and $(\rho,\theta,z)$ are the cylindrical coordinates. This means essentially that the field at $\rho=0$ is zero and that it is proportional to $\rho$ inside the conductor and inversely proportional to $\rho$ on the outside. How can we get this from the solution to Poisson's equation for a finite-radius wire?
The other thing that troubles me with the solution to Poisson's equation (second equation) is the value of the integrand when $\vec{r}=\vec{r'}$, but outside the wire (thus where J=0). This means we get a 0/0 integrand for each $\vec{r}$ outside the wire, which numerically gives NaN for the whole integral. Is this a problem analytically? because this contribution might (should) be 0, probably by using L'Hopital's rule (I guess).
M.
[1] Smythe,W.R., "Static and dynamic electricity", McGraw-Hill, 1968.
I have questions regarding Biot-Savart law. From [1], it is shown that the equation (Biot-Savart) is derived from the solution to Poisson's equation (assuming here div A=0)
\begin{equation}
\vec{\nabla}^2 \vec{A} = -\mu \vec{J}
\end{equation}
which is
\begin{equation}
\vec{A}(\vec{r}) =\frac{\mu}{4\pi}\int_V{\frac{\vec{J}{\rm d}^3r'}{\left|\vec{r}-\vec{r'}\right|}}
\end{equation}
where $\vec{r}$ is the position where $\vec{A}$ is evaluated and $\vec{r'}$ is the position where the integral is evaluated.
The first thing that troubles me is the singularity $\left|\vec{r}-\vec{r'}\right|$ when we evaluate the field at the point of integration. For a wire of finite radius, this means that the $\vec{A}$ field inside the conductor is infinite (or am I missing something?). If so, why in books on electromagnetics do we usually replace the conductor by an equivalent filamentary current $I=\vec{J}\cdot{\rm d}\vec{s}$? The field calculated inside the conductor will be different. This can be seen from the solution for the $\vec{B}$ field by using Ampere's equation (for the infinitely-long finite-radius wire)
\begin{equation}
B_{\theta}=\frac{\mu I}{2 \pi \rho}
\end{equation}outside the wire
\begin{equation}
B_{\theta}=\frac{\mu I \rho}{2 \pi R_{wire}^2}
\end{equation} inside the wire
where $R_{wire}$ is the cross-section radius and $(\rho,\theta,z)$ are the cylindrical coordinates. This means essentially that the field at $\rho=0$ is zero and that it is proportional to $\rho$ inside the conductor and inversely proportional to $\rho$ on the outside. How can we get this from the solution to Poisson's equation for a finite-radius wire?
The other thing that troubles me with the solution to Poisson's equation (second equation) is the value of the integrand when $\vec{r}=\vec{r'}$, but outside the wire (thus where J=0). This means we get a 0/0 integrand for each $\vec{r}$ outside the wire, which numerically gives NaN for the whole integral. Is this a problem analytically? because this contribution might (should) be 0, probably by using L'Hopital's rule (I guess).
M.
[1] Smythe,W.R., "Static and dynamic electricity", McGraw-Hill, 1968.
