Let me explain in details. Consider a region in space with no free charges and no free currents, so that the charge density is $\rho=0$ and the volumetric current is $\vec{J}=\vec{0} \text{ }$ in the entire space. Let $\vec{B}=\vec{B}(\vec{x},t) \text{ }$ be the magnetic field in the space. Suppose that $\vec{B} \text{ }$ is uniform in space, that is, suppose that $\vec{B} \text{ }$ is only a function of time: $\vec{B}(\vec{x},t)=\vec{B}(t) \text{ }$. Moreover, suppose that $\vec{B}(t)=at\hat{e}_z$ for some constant $a$. Notice that although $\vec{B} \text{ }$ is uniform, $\vec{B} \text{ }$ varies with time. Under these circumstances, what is the electric field generated by $\vec{B} \text{ }$ in the entire space?
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Im sorry, but the answer cannot be determined from the given information. We know $ E $ is constant in time, that the curl of $E$ is constant in time and space, but that is not enough to determine $E$. I can write down an infinite number of solutions which satisfy those requirements. You need to specify some additional boundary conditions. |
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$\nabla\times B = \mu_0 J + \mu_0\epsilon_0 \frac{\partial E}{\partial t}$ Since the magnetic field is constant in position, $\nabla \times B=0$, and so, since $J=0$, you have $E$ doesn't change (and stays zero). If the obvious implications of what I wrote above isn't clear to the reader, let me spell it out here. The OP asks what electric field is "generated". This means that I am not to assume that there's some previously existing electric field. So at time $t=0$, we assume $\vec{E} = 0$ everywhere. But the above shows that $\vec{E}$ stays constant. Therefore, $\vec{E}$ remains zero, and no electric field is generated. As a general comment that the reader will find useful for all sorts of coupled sets of partial differential equations describing waves, if one is interested in the evolution over time of a solution, one should pay attention to the partial derivatives with respect to time and solve for them. This is a different form of Maxwell's equations. This is the sort of thing you have to do if you are going to write a computer program which approximates the evolution of a solution to Maxwell's equations. That is, solve for $\partial E/\partial t$ and $\partial B/\partial t$. |
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The unique axisymmetric field is $$ E_x(t) = - {ay/2} $$ $$ E_y(t) = {ax/2} $$ Which obviously satisfies Maxwell's equations. This would be the electric field generated in a circular solenoid centered at the origin. Note that the field has a center, it breaks the naive translation invariance. You can add any electrostatic field $E'$ to the electric field above, and the electrostatic field to add is determined by the boundary conditions. Using a constant electrostatic field, you shift the center of rotational invariance from the origin to another point. Using the general 2+1 quadratic electrostatic potential (the real and imaginary parts of the complex squaring function) $$ \phi(x,y) = C xy + D (y^2-x^2)$$ You get a more general solution $$ E_x = ({-a\over 2} + C) y - D x$$ $$ E_y = ({a\over 2} + C) x - D y $$ Setting D=0 and C= a/2, you get the field which is correct for an infinitely long elliptical solenoid along the z-axis, centered at the origin (two parallel plates of current in opposite directions). These cases are determined by symmetry, to specify the electrostatic field for a general configuration requires more information than is given, as user1631 said, in particular, the boundary conditions for E. |
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