Electromagnetic field

Electric and magnetic fields in vacuum.

(a)Consider a region of space free from matter and from magnetic fields.Show that it cannot be permeated by an electric field of the form:

\vec{E}(x, y, z)=\frac{A}{2}(y \hat{x}-x \hat{y}+z \hat{z})

\text { where } A \text { is a constant value different from zero and with units } \mathrm{V} / \mathrm{m}^{2} \text {. }

Now, consider that a magnetic field permeates the region of space in(a). Write an expression representing a time-varying magnetic field B(r,y, z, t) that renders the vector field proposed in (a) suitable to describe an electric field. Check and demonstrate that the expression you found is a suitable representation of a magnetic field.

Hence or otherwise, show that:

\vec{B}=K x \hat{x}+K y \hat{y}+A t \hat{z}

\text { is not an adequate expression for a magnetic field for } K \neq 0 \mathrm{~T} / \mathrm{m}^{2} \text {. }

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