First slide

Magnetic field due to current element

Question

A particle is moving with velocity $\bar{V} = \hat{i} + 3 \hat{j}$ and it produces an electric field at a point given by $\vec{E} = 2 \hat{k}$ . It will produce magnetic field at that point equal to (all quantities are in S.I units)

Difficult

A

$(6 \hat{i} - 2 \hat{j}) μ_{o} ε_{0}$
B

$(6 \hat{i} + 2 \hat{j}) μ_{o} ε_{0}$
C

$z e r o$
D

cannot be determined from the given data

Solution

$\vec{E} = \frac{1}{4 π ε_{0}} \frac{q}{r^{2}} \hat{r} = 2 \hat{k} ∴ W e c a n s a y t h a t d i r e c t i o n o f \vec{r} i s a l o n g \hat{k} a n d \frac{1}{4 π ε_{0}} \frac{q}{r^{2}} = 2 N o w, a s p e r B i o t - S a v a r t L a w \vec{B} = \frac{μ_{0}}{4 π} \frac{q \vec{v} \times \vec{r}}{r^{2}} = μ_{0} ε_{0} [\frac{q}{4 π ε_{0} r^{2}}] (\hat{i} + 3 \hat{j}) \times \hat{k} = μ_{0} ε_{0} 2 (- \hat{j} + 3 \hat{i}) = (6 \hat{i} - 2 \hat{j}) μ_{0} ε_{0}$

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