Consider a region of a magnetic field varying uniformly with space such that change of magnetic field is along z-axis of a cylindrical coordinate system. Now, this will obviously induce an electric field. I tried deducing tthe magnitude and direction of electric field using Maxwell's equations as follows:
1. Since no charges are present, Gauss Law gives $E_r$ = 0
2. As $\nabla \times \bf E$ is perpendicular to $\bf E$, $E_z = 0$
Thus $$\frac{\partial( r E_\phi)}{r \partial r} = \frac{\partial B}{\partial t}$$ Integrating, we get, $$E = \frac{1}{2}r\frac{\partial B}{\partial t}$$
However, this expression seems to depend on my choice of the origin of my cylindrical coordinate system which is clearly not possible. My question is, am I doing something wrong? If not, what is the position from which I have to measure $r$?
