The electric field of a plane polarized electromagnetic wave in free space at time $$t = 0$$ is given by the expression $$\vec{E}(x, y) = 10\hat{j}\cos(6x + 8z)$$. The magnetic field $$\vec{B}(x, z, t)$$ is given by ($$c$$ is the velocity of light.)

We have the electric-field vector at time $$t = 0$$ written as

$$ \vec E(x,z,0)=10\,\hat{\jmath}\;\cos(6x+8z). $$

In a monochromatic plane wave the space-time dependence is of the form $$\cos(\vec k\!\cdot\!\vec r-\omega t)$$. Comparing the given argument $$6x+8z$$ with $$\vec k\!\cdot\!\vec r$$, we identify the wave-vector

$$ \vec k = 6\,\hat{\imath}+8\,\hat{k}. $$

The magnitude of this vector is calculated step by step:

$$ |\vec k| = \sqrt{6^{2}+8^{2}}=\sqrt{36+64}= \sqrt{100}=10. $$

For an electromagnetic wave in free space the angular frequency and the wave-vector are related by the formula $$\omega = c\,|\vec k|$$. Substituting the value of $$|\vec k|$$ we get

$$ \omega = c \times 10 = 10\,c. $$

Therefore the complete electric field, including its time variation, is

$$ \vec E(x,z,t)=10\,\hat{\jmath}\;\cos(6x+8z-\omega t)=10\,\hat{\jmath}\;\cos(6x+8z-10ct). $$

For a plane wave propagating in the direction $$\hat{k}=\dfrac{\vec k}{|\vec k|}$$, Maxwell’s equations give the magnetic field through the well-known vector relation

$$ \vec B=\dfrac{1}{c}\,\hat{k}\times\vec E. $$

First we obtain the unit vector $$\hat{k}$$ by dividing each component of $$\vec k$$ by its magnitude:

$$ \hat{k}= \dfrac{1}{10}\,(6\,\hat{\imath}+8\,\hat{k}) = 0.6\,\hat{\imath}+0.8\,\hat{k}. $$

Now we evaluate the cross product $$\hat{k}\times\vec E$$ algebraically, keeping every step visible.

$$ \begin{aligned} \hat{k}\times\vec E &= (0.6\,\hat{\imath}+0.8\,\hat{k})\times\bigl(10\,\hat{\jmath}\bigr) \\ &= 0.6\,( \hat{\imath}\times10\hat{\jmath} ) + 0.8\,( \hat{k}\times10\hat{\jmath} ) \\ &= 0.6\times10\;(\hat{\imath}\times\hat{\jmath}) + 0.8\times10\;(\hat{k}\times\hat{\jmath}) \\ &= 6\,\hat{k} + 8\,(-\hat{\imath}) \\ &= 6\,\hat{k}-8\,\hat{\imath}. \end{aligned} $$

Dividing this by $$c$$ as required by the formula, we get the magnetic-field amplitude

$$ \dfrac{1}{c}\,\bigl(6\,\hat{k}-8\,\hat{\imath}\bigr). $$

The phase factor of the magnetic field is identical to that of the electric field, namely $$\cos(6x+8z-10ct)$$. Putting everything together, the complete magnetic field is

$$ \vec B(x,z,t)=\dfrac{1}{c}\,(6\,\hat{k}-8\,\hat{\imath})\;\cos(6x+8z-10ct). $$

This expression coincides exactly with Option D.

Hence, the correct answer is Option D.