Chapter 1: Problem 33
If you have some experience in electromagnetism and with vector calculus, prove that the magnetic forces, \(\mathbf{F}_{12}\) and \(\mathbf{F}_{21}\), between two steady current loops obey Newton's third law. [Hints: Let the two currents be \(I_{1}\) and \(I_{2}\) and let typical points on the two loops be \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\). If \(d \mathbf{r}_{1}\) and \(d \mathbf{r}_{2}\) are short segments of the loops, then according to the Biot- Savart law, the force on \(d \mathbf{r}_{1}\) due to \(d \mathbf{r}_{2}\) is $$\frac{\mu_{0}}{4 \pi} \frac{I_{1} I_{2}}{s^{2}} d \mathbf{r}_{1} \times\left(d \mathbf{r}_{2} \times \hat{\mathbf{s}}\right)$$ where \(\mathbf{s}=\mathbf{r}_{1}-\mathbf{r}_{2} .\) The force \(\mathbf{F}_{12}\) is found by integrating this around both loops. You will need to use the " \(B A C-C A B\) " rule to simplify the triple product.]
Short Answer
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Key Concepts
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