Question

If vector field A = (x + 3y) î + (y – 2z)j + (x + mz)k is solenoidal, then the value of m is
(A) 2
(8)3
(C)-2
(D)0​

Answer 1

If vector field $\overrightarrow{A}=(x+3y)\hat{i}+(y-2z)\hat{j}+(x+mz)\hat{k}$ is solenoidal, then the value of $m$ is $(-2)$.

Correct option: (C) $-2$

Step-by-step explanation:

Given that, the vector field $\overrightarrow{A}$ is solenoidal. Then its divergence is $0$.

$\Rightarrow div\:(\overrightarrow{A})=0$

That is, $\overrightarrow{\nabla}.\overrightarrow{A}=0$

$\Rightarrow (\hat{i}\dfrac{\partial}{\partial x}+\hat{j}\dfrac{\partial}{\partial y}+\hat{k}\dfrac{\partial}{\partial z}).\{(x+3y)\hat{i}\\+(y-2z)\hat{j}+(x+mz)\hat{k}\}=0$

$\Rightarrow \dfrac{\partial}{\partial x}(x+3y)+\dfrac{\partial}{\partial y}(y-2z)\\+\dfrac{\partial}{\partial z}(x+mz)=0$

$\Rightarrow 1+1+m=0$

$\Rightarrow m=-2$

Thus the value of $m$ is $(-2)$.