Electromagnetism | Part IB, 2004

The vector potential due to a steady current density J\mathbf{J} is given by

A(r)=μ04πJ(r)rrd3r\mathbf{A}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d^{3} \mathbf{r}^{\prime}

where you may assume that J\mathbf{J} extends only over a finite region of space. Use ()(*) to derive the Biot-Savart law

B(r)=μ04πJ(r)×(rr)rr3d3r\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right) \times\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} d^{3} \mathbf{r}^{\prime}

A circular loop of wire of radius aa carries a current II. Take Cartesian coordinates with the origin at the centre of the loop and the zz-axis normal to the loop. Use the BiotSavart law to show that on the zz-axis the magnetic field is in the axial direction and of magnitude

B=μ0Ia22(z2+a2)3/2B=\frac{\mu_{0} I a^{2}}{2\left(z^{2}+a^{2}\right)^{3 / 2}}

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