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Magnetism and Matter question

2021 · Q160
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Magnetism and Matter question

2021 · Q160

NEETPhysicsMagnetism and MatterMCQ+4 / −1
In the product

F→=q(v→×B→)\overrightarrow F = q\left( {\overrightarrow v \times \overrightarrow B } \right)F=q(v×B)

=qv→×(Bi^+Bj^+B0k^)= q\overrightarrow v \times \left( {B\widehat i + B\widehat j + {B_0}\widehat k} \right)=qv×(Bi+Bj​+B0​k)

For q = 1 and v→=2i^+4j^+6k^\overrightarrow v = 2\widehat i + 4\widehat j + 6\widehat kv=2i+4j​+6k and F→=4i^−20j^+12k^\overrightarrow F = 4\widehat i - 20\widehat j + 12\widehat kF=4i−20j​+12k

What will be the complete expression for B→\overrightarrow BB ?
  1. A
    6i^+6j^−8k^6\widehat i + 6\widehat j - 8\widehat k6i+6j​−8k
  2. B
    −8i^−8j^−6k^- 8\widehat i - 8\widehat j - 6\widehat k−8i−8j​−6k
  3. C
    −6i^−6j^−8k^- 6\widehat i - 6\widehat j - 8\widehat k−6i−6j​−8k
  4. D
    8i^+8j^−6k^8\widehat i + 8\widehat j - 6\widehat k8i+8j​−6k
View written solutionFree

Correct answer: C

Given q = 1 and v→=2i^+4j^+6k^\overrightarrow v = 2\widehat i + 4\widehat j + 6\widehat kv=2i+4j​+6k and F→=4i^−20j^+12k^\overrightarrow F = 4\widehat i - 20\widehat j + 12\widehat kF=4i−20j​+12k

Also given, F→=q(v→×B→)\overrightarrow F = q\left( {\overrightarrow v \times \overrightarrow B } \right)F=q(v×B)

=qv→×(Bi^+Bj^+B0k^) = q\overrightarrow v \times \left( {B\widehat i + B\widehat j + {B_0}\widehat k} \right)=qv×(Bi+Bj​+B0​k)

⇒\Rightarrow⇒ (4i^−20j^+12k^)=−1×[(2i^+4j^+6k^)×(Bi^+Bj^+B0k^)]\left( {4\widehat i - 20\widehat j + 12\widehat k} \right) = - 1 \times \left[ {\left( {2\widehat i + 4\widehat j + 6\widehat k} \right) \times \left( {B\widehat i + B\widehat j + {B_0}\widehat k} \right)} \right](4i−20j​+12k)=−1×[(2i+4j​+6k)×(Bi+Bj​+B0​k)]

Thus, calculating values of RHS,

$$\overrightarrow v \times \overrightarrow B = \left| {\begin{matrix} {\widehat i} & {\widehat j} & {\widehat k} \ 2 & 4 & 6 \ B & B & {{B_0}} \

\end{matrix} } \right|<br><br>= ${\widehat i}$(4B<sub>0</sub> - 6B) - ${\widehat j}$(2B<sub>0</sub> - 6B) + ${\widehat k}$(2B - 4B) <br><br>Comparing L.H.S and R.H.S, <br><br>4B<sub>0</sub> – 6B = 4 .....(1) <br><br>–(2B<sub>0</sub> – 6B)= –20 .....(2) <br><br>2B – 4B = 12 $ \Rightarrow $ B = –6 .....(3) <br><br>From (2) and (3) B = –6 and B<sub>0</sub> = –8 <br><br>$ \therefore $ $\overrightarrow B $ = - 6\widehat i - 6\widehat j - 8\widehat k$$

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