In a certain region static electric and magnetic fields exist. The magnetic field is given by $$\vec{B} = B_0(\hat{i} + 2\hat{j} - 4\hat{k})$$. If a test charge moving with a velocity $$\vec{v} = v_0(3\hat{i} - \hat{j} + 2\hat{k})$$ experiences no force in that region, then the electric field in the region, in SI units, is:

For a charge moving in simultaneous electric and magnetic fields, the Lorentz force formula states

$$\vec{F}=q\left(\vec{E}+\,\vec{v}\times\vec{B}\right).$$

Here the problem tells us that the charge “experiences no force,” so the net force vector must be zero. Hence, setting $$\vec{F}=0$$ gives

$$\vec{E}+\,\vec{v}\times\vec{B}=0.$$

Re-arranging, we obtain an explicit expression for the electric field:

$$\vec{E}=-\,\vec{v}\times\vec{B}.$$

Now we substitute the given velocity and magnetic-field vectors. They are

$$\vec{v}=v_0\left(3\hat{i}-\hat{j}+2\hat{k}\right),\qquad \vec{B}=B_0\left(\hat{i}+2\hat{j}-4\hat{k}\right).$$

To evaluate the cross product $$\vec{v}\times\vec{B}$$ we write the determinant in component form:

$$ \vec{v}\times\vec{B}= \begin{vmatrix} \hat{i}&\hat{j}&\hat{k}\\[4pt] 3v_0&-v_0&2v_0\\[4pt] B_0&2B_0&-4B_0 \end{vmatrix}. $$

Expanding this determinant component by component:

• The $$\hat{i}$$ component is $$ \hat{i}\Big[(-v_0)(-4B_0)-(2v_0)(2B_0)\Big] =\hat{i}\Big[4v_0B_0-4v_0B_0\Big] =0\,\hat{i}. $$

• The $$\hat{j}$$ component carries a minus sign in the determinant expansion, so

$$ -\hat{j}\Big[(3v_0)(-4B_0)-(2v_0)(B_0)\Big] =-\hat{j}\Big[-12v_0B_0-2v_0B_0\Big] =-\hat{j}\big[-14v_0B_0\big] =14v_0B_0\,\hat{j}. $$

• The $$\hat{k}$$ component is

$$ \hat{k}\Big[(3v_0)(2B_0)-(-v_0)(B_0)\Big] =\hat{k}\Big[6v_0B_0+v_0B_0\Big] =7v_0B_0\,\hat{k}. $$

Collecting all three components, the cross product becomes

$$ \vec{v}\times\vec{B}=v_0B_0\big(0\,\hat{i}+14\,\hat{j}+7\,\hat{k}\big) =v_0B_0\left(14\hat{j}+7\hat{k}\right). $$

Finally, substituting this result into $$\vec{E}=-\,\vec{v}\times\vec{B}$$ we get

$$ \vec{E}=-v_0B_0\left(14\hat{j}+7\hat{k}\right). $$

Hence, the correct answer is Option D.