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A small current element of length $d \ell$ and carrying current is placed at $(1,1,0)$ and is carrying current in ' $+ z$ ' direction. If magnetic field at origin be $\overrightarrow{ B }_1$ and at point $(2,2,0)$ be $\overrightarrow{ B }_2$ then
$\overrightarrow{ B }_1=\overrightarrow{ B }_2$
$\left|\overrightarrow{ B }_1\right|=\left|2 \overrightarrow{ B }_2\right|$
$\overrightarrow{ B }_1=-\overrightarrow{ B }_2$
$\overrightarrow{ B }_1=-2 \overrightarrow{ B }_2$
Solution
(c)
$\vec{B}=\frac{\mu_0}{4 \pi} \frac{\times \vec{r}}{r^3}$ for $B_1 \;\;\vec{r}=(-\hat{i}-\hat{j})$
$\therefore \vec{B}_1=\frac{\mu_0}{4 \pi} \frac{i}{2 \sqrt{2}} \hat{k} \times(-\hat{i}-\hat{j}) \ldots .(1)$
for $B_2\;\; \vec{r}=\hat{i}+\hat{j}$
$\vec{B}_2=\frac{\mu_0}{4 \pi} \frac{i \hat{k} \times(\hat{i}+\hat{j})}{2 \sqrt{2}} \ldots \ldots(2)$
from (1) and (2)
$\vec{B}_t=-\vec{B}_2$ and $\left|\vec{B}_1\right|=\left|\vec{B}_2\right|$
