Learning Introductory Physics with Activities

For a continuous current density, you can use the Chop-Multiply-Add strategy to construct and evaluate an integral for the magnetic field, similar to the strategy in Section 9.9. The overall strategy for the magnetic field above the middle of a strip of total current \(I\) that is infinite in the \(x\)-direction (with width \(L\) in the \(y\)-direction) is given below.

Chop: You can start by chopping up the strip into wires. Each wire carries infinitesimal current \(\vec{dI} = \frac{I}{L} dy \hat{x}\text{,}\) where the \(dy\) represents an infinitesimal length.

Multiply: Once you know the current carried by each small wire \(dI\text{,}\) you multiply to find the magnetic field created by that small wire: \(d\vec{B} = \frac{\mu_o}{4\pi}\frac{\vec{dI} \times \hat{r}}{r^2} = \frac{\mu_o I}{4\pi L}\frac{\hat{x} \times \hat{r}}{r^2}dy\text{.}\) Here \(r\) is the distance from the location of \(\vec{dI}\) to the location where you are evaluating the magnetic field, and \(\hat{r}\) is the unit vector that points in this direction. It can be highly useful to draw a diagram of the current, including an example \(\vec{dI}\text{,}\) and label \(r\) and \(\hat{r}\) so they can be easily written in terms of your chosen coordinates. This is especially useful for calculating the cross product.