Potentials Based Field Calculation

This problem requires calculating the electric field ($\vec{E}$) and magnetic field ($\vec{B}$) from given scalar ($\phi$) and vector ($\vec{A}$) potentials.

Given Potentials

• Scalar potential: $\phi(z,t) = 0$
• Vector potential: $\vec{A}(z,t) = \hat{i} t z$. (Note: Interpreting the unconventional notation $\hat it{z}$ as $\hat{i} t z$)

Electromagnetic Field Formulas

The fundamental relations between fields and potentials are:

• Electric Field: $\vec{E} = -\nabla \phi - \frac{\partial \vec{A}}{\partial t}$
• Magnetic Field: $\vec{B} = \nabla \times \vec{A}$

Electric Field Calculation

Using the given potentials:

Since $\phi = 0$, its gradient is zero: $\nabla \phi = 0$.

The formula simplifies to $\vec{E} = - \frac{\partial \vec{A}}{\partial t}$.

Calculate the partial derivative of $\vec{A}$ with respect to time $t$:

$ \frac{\partial \vec{A}}{\partial t} = \frac{\partial}{\partial t}(\hat{i} t z) = \hat{i} z $

Therefore, the electric field is:

$ \vec{E} = - (\hat{i} z) = -\hat{i} z $

Magnetic Field Calculation

Calculate the curl of the vector potential $\vec{A}$:

$ \vec{B} = \nabla \times \vec{A} = \nabla \times (\hat{i} t z) $

Using the curl formula in Cartesian coordinates:

$ \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ tz & 0 & 0 \end{vmatrix} $

Expanding the determinant yields:

$ \vec{B} = \hat{i} \left( \frac{\partial}{\partial y}(0) - \frac{\partial}{\partial z}(0) \right) - \hat{j} \left( \frac{\partial}{\partial x}(0) - \frac{\partial}{\partial z}(tz) \right) + \hat{k} \left( \frac{\partial}{\partial x}(0) - \frac{\partial}{\partial y}(tz) \right) $

$ \vec{B} = 0\hat{i} - \hat{j}(0 - t) + 0\hat{k} = \hat{j} t $

Result Matches Option

The calculated fields are $\vec{E} = -\hat{i} z$ and $\vec{B} = \hat{j} t$.

This corresponds to Option D.