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Question
The magnetic field at the origin due to a current element \[i d \vec{l}\] placed at a position \[\vec{r}\] is
(a)\[\frac{\mu_0 i}{4\pi}\frac{d \vec{l} \times \vec{r}}{r^3}\]
(b) \[- \frac{\mu_0 i}{4\pi}\frac{\vec{r} \times d \vec{l}}{r^3}\]
(c) \[\frac{\mu_0 i}{4\pi}\frac{\vec{r} \times d \vec{l}}{r^3}\]
(d) \[- \frac{\mu_0 i}{4\pi}\frac{d \vec{l} \times \vec{r}}{r^3}\]
Solution
(a) \[\frac{\mu_0 i}{4\pi}\frac{d \vec{l} \times \vec{r}}{r^3}\]
(b) \[- \frac{\mu_0 i}{4\pi}\frac{\vec{r} \times d \vec{l}}{r^3}\]
The magnetic field at the origin due to current element \[i d \vec{l}\] placed at a position \[\vec{r}\] is given by \[d \vec{B} = \frac{\mu_o}{4\pi}i\frac{\vec{r} \times d \vec{l}}{r^3}\]
According to the cross product property,
\[\vec{A} \times \vec{B} = - \vec{B} \times \vec{A} \]
\[ \Rightarrow d \vec{B} = - \frac{\mu_o}{4\pi}i\frac{\vec{r} \times \vec{dl}}{r^3}\]
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