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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 $r$?

Srish Dutta
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  • Uniform magnetic field is too few assumptions, it does not determine electric field uniquely. You need to assume more, e.g. assume some boundary condition on the magnetic field or electric field. If the magnetic field is due to finite cylindrical magnet and thus falls off with distance, this defines the center of symmetry of both fields and you can find $\mathbf E$ from $\mathbf B$. – Ján Lalinský Aug 12 '23 at 16:17
  • @JánLalinský can you please explain further – Srish Dutta Aug 13 '23 at 03:29
  • https://physics.stackexchange.com/questions/494553/electric-field-due-to-changing-uniform-magnetic-field https://physics.stackexchange.com/questions/494426/electric-field-in-a-uniform-time-varying-magnetic-field – Ján Lalinský Aug 13 '23 at 11:04

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