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) \[- \frac{\mu_0 i}{4\pi}\frac{d \vec{l} \times \vec{r}}{r^3}\] 

Answer 1

(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,

\[\vec{A} \times \vec{B} = - \vec{B} \times \vec{A} \]
\[ \Rightarrow d \vec{B} = - \frac{\mu_o}{4\pi}i\frac{\vec{r} \times \vec{dl}}{r^3}\]