JEE MAIN - Physics (2017 - 8th April Morning Slot - No. 10)
In a certain region static electric and magnetic fields exist. The magnetic field is given by $$\overrightarrow B = {B_0}\left( {\widehat i + 2\widehat j - 4\widehat k} \right)$$ . If a test charge moving with a velocity $$\overrightarrow \upsilon = {\upsilon _0}\left( {3\widehat i - \widehat j + 2\widehat k} \right)$$ experiences no force in that region, then the electric field in the region, in SI units, is :
$$\overrightarrow E = - {\upsilon _0}\,{B_0}\left( {3\widehat i - 2\widehat j - 4\widehat k} \right)$$
$$\overrightarrow E = - {\upsilon _0}\,{B_0}\left( {\widehat i + \widehat j + 7\widehat k} \right)$$
$$\overrightarrow E = {\upsilon _0}\,{B_0}\left( {14\widehat j + 7\widehat k} \right)$$
$$\overrightarrow E = - {\upsilon _0}\,{B_0}\left( {14\widehat j + 7\widehat k} \right)$$
Explanation
Here test charge experience no net force, So, sum of electric and magnetic field is zero.
$$\therefore\,\,\,$$ Fe + Fm = 0
$$\therefore\,\,\,$$ Fe = $$-$$ q ($$\overrightarrow v $$ $$ \times $$ $$\overrightarrow B)$$
= $$-$$ qB0 $$\upsilon $$0 [(3$$\widehat i$$ $$-$$ $$\widehat j$$ + 2$$\widehat k$$) $$ \times $$ ($$\widehat i$$ + 2$$\widehat j$$ $$-$$ 4$$\widehat k$$)]
= $$-$$ q$$\upsilon $$0 B0 (14$$\widehat j$$ + 7$$\widehat k$$)
Electric field produced by the charge q,
$$\overrightarrow E $$ = $${{\overrightarrow {{F_e}} } \over q}$$
= $${{ - q{\upsilon _0}{B_0}\left( {14\widehat j + 7\widehat k} \right)} \over q}$$
= $$-$$ $$\upsilon $$0 B0 (14 $${\widehat j}$$ + 7 $${\widehat k}$$)
$$\therefore\,\,\,$$ Fe + Fm = 0
$$\therefore\,\,\,$$ Fe = $$-$$ q ($$\overrightarrow v $$ $$ \times $$ $$\overrightarrow B)$$
= $$-$$ qB0 $$\upsilon $$0 [(3$$\widehat i$$ $$-$$ $$\widehat j$$ + 2$$\widehat k$$) $$ \times $$ ($$\widehat i$$ + 2$$\widehat j$$ $$-$$ 4$$\widehat k$$)]
= $$-$$ q$$\upsilon $$0 B0 (14$$\widehat j$$ + 7$$\widehat k$$)
Electric field produced by the charge q,
$$\overrightarrow E $$ = $${{\overrightarrow {{F_e}} } \over q}$$
= $${{ - q{\upsilon _0}{B_0}\left( {14\widehat j + 7\widehat k} \right)} \over q}$$
= $$-$$ $$\upsilon $$0 B0 (14 $${\widehat j}$$ + 7 $${\widehat k}$$)
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