the magnetic field of a thin long wire.We will directly use the Biot-Savart law to calculate the magnetic field of a thin wire via integration. The problem shall provide us with some intuition of Biot-Savart which we will also derive.

Problem Statement

The magnetic field of a thin and infinite wire. In The Magnetic Field of an Infinite Wire we discuss the field of a not necessarily thin wire with a constant current \(j_{0}\mathbf{e}_{z}\) and discuss generalizations \(j_{0}\rightarrow j\left(\rho\right)\). We made our calculations directly from the differential form of Ampère's law. If we assume a very thin wire, i.e. a current of the form\[\mathbf{j}\left(\mathbf{r}\right)    =    I\delta\left(\rho\right)\mathbf{e}_{z}\]in cylindrical coordinates, we may also calculate the magnetic field by direct integration via Biot-Savart's law,\[\mathbf{B}\left(\mathbf{r}\right)    =    \frac{\mu_{0}I}{4\pi}\int\frac{d\mathbf{s}\times\left(\mathbf{r}-\mathbf{r}\left(s\right)\right)}{\left|\mathbf{r}-\mathbf{r}\left(s\right)\right|^{3}}\ ,\]where the integral is over the curve of the thin wire.

  • Calculate the magnetic field \(\mathbf{B}\left(\mathbf{r}\right)\) for the thin wire, infinitely extended in \(z\) using Biot-Savart law.
  • Extra: Derive the Biot-Savart law from the connection of the vector potential to the current.

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