MCQ
The magnetic field at the origin due to a current element $\text{i}\text{d}\ \vec{\text{l}}$ placed at a position $\overrightarrow{\text{r}}$ is:
  1. $\frac{\mu_0\text{i}}{4\pi}\frac{\text{d}\overrightarrow{\text{l}}\times\overrightarrow{\text{r}}}{\text{r}^3}$
  2. $-\frac{\mu_0\text{i}}{4\pi}\frac{\overrightarrow{\text{r}}\times​​\text{d}\overrightarrow{\text{l}}}{\text{r}^3}$
  3. $\frac{\mu_0\text{i}}{4\pi}\frac{\overrightarrow{\text{r}}\times​​\text{d}\overrightarrow{\text{l}}}{\text{r}^3}$
  4. $-\frac{\mu_0\text{i}}{4\pi}\frac{\text{d}\overrightarrow{\text{l}}\times\overrightarrow{\text{r}}}{\text{r}^3}$
  • A
    Only $A$
  • $A$ and $B$
  • C
    Only $C$
  • D
    $B$ and $D$

Answer

Correct option: B.
$A$ and $B$
The magnetic field at the origin due to current element $\text{i}\text{d}\ \vec{\text{l}}$ placed at a position $\overrightarrow{\text{r}}$ is given by,
$\text{d}\overrightarrow{\text{B}}=\frac{\mu_0\text{i}}{4\pi}\frac{\overrightarrow{\text{r}}\times\text{d}\overrightarrow{\text{l}}}{\text{r}^3}$
According to the cross product property,
$\overrightarrow{\text{A}}\times\overrightarrow{\text{B}}=-\overrightarrow{\text{B}}\times\overrightarrow{\text{A}}$
$\Rightarrow\text{d}\overrightarrow{\text{B}}=-\frac{\mu_0\text{i}}{4\pi}\frac{\overrightarrow{\text{r}}\times\text{d}\overrightarrow{\text{l}}}{\text{r}^3}$

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