Question

The divergence of a magnetic vector field B\vec{B} must be zero everywhere. Which of the following vector fields cannot be a magnetic vector field?

B(x,y,z)=zi+yj+xk\vec{B} (x, y, z) = −z\vec{i} + y\vec{j} + x\vec{k}

Solution

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Answered 1 year ago
Answered 1 year ago
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Let's find the divergence of the given vector field:

divB=div(zi+yj+xk)=x(z)+y(y)+z(x)=0+1+0=1\begin{aligned} div\vec{B}&=div(-z\vec{i}+y\vec{j}+x\vec{k})\\ &=\dfrac{\partial}{\partial x}(-z)+\dfrac{\partial}{\partial y}(y)+\dfrac{\partial}{\partial z}(x)\\ &=0+1+0\\ &=1 \end{aligned}

Since divB0div\vec{B}\ne0, the given vector field cannot be a magnetic vector field.

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