A charged particle is projected in a plane perpendicular to a uniform magnetic field. The area bounded by the path described by the particle is proportional to

The magnetic moments associated with two closely wound circular coils $A$ and $B$ of radius $r_A=10 cm$ and $r_B=20 cm$ respectively are equal if: (Where $N _A, I _{ A }$ and $N _B, I _{ B }$ are number of turn and current of $A$ and $B$ respectively)

A particle of mass $M$ and charge $Q$ moving with velocity $\mathop v\limits^ \to $ describes a circular path of radius $R$ when subjected to a uniform transverse magnetic field of induction $B$. The work done by the field when the particle completes one full circle is

A particle of mass $m,$ charge $Q$ and kinetic energy $K$ enters a transverse uniform magnetic field of induction $B.$ After $3$ $seconds$ the kinetic energy of the particle will be .......$K$

A proton of velocity $\left( {3\hat i + 2\hat j} \right)\,ms^{-1}$ enters a magnetic field of  $(2\hat j + 3\hat k)\, tesla$. The acceleration produced in the proton is (charge to mass ratio of proton $= 0.96 \times10^8\,Ckg^{-1}$)

In the product

$\overrightarrow{\mathrm{F}} =\mathrm{q}(\vec{v} \times \overrightarrow{\mathrm{B}})$

$=\mathrm{q} \vec{v} \times\left(\mathrm{B} \hat{i}+\mathrm{B} \hat{j}+\mathrm{B}_{0} \hat{k}\right)$

For $\mathrm{q}=1$ and $\vec{v}=2 \hat{i}+4 \hat{j}+6 \hat{k}$ and

$\overrightarrow{\mathrm{F}}=4 \hat{i}-20 \hat{j}+12 \hat{k}$

What will be the complete expression for $\vec{B}$ ?