Does a linearly time varying B create changing E?

[Shreya]

Homework Statement

I know that a time varying magnetic field, creates a curling electric field around it. My question is this: If the time variance is linear, the curl of electric field is constant. Doesn't that imply electric field is not varying with time? So, no curling magnetic field is generated?

Relevant Equations

Maxwell's 4 laws

As accelerated charges produces time varying electric field which produces time varying magnetic fields and so on. I know there is something wrong with my argument, I can't point it out. Please be kind to help.

 

[pasmith]

You have $\mathbf{B} = \mathbf{b}(\mathbf{x})t$ where $\nabla \cdot \mathbf{b} = 0$. Hence in the absence of sources, $$(\nabla \times \mathbf{b})t = \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}$$ so that $$\mathbf{E} = \frac{c^2t^2}{2} (\nabla \times \mathbf{b}) + \mathbf{E}_0(\mathbf{x}).$$ From $$\nabla \times \mathbf{E} = -\mathbf{b}$$ we find $$\frac{c^2t^2}{2}\nabla \times( \nabla\times \mathbf{b}) = - \nabla \times \mathbf{E}_0 - \mathbf{b}.$$ Now the right hand side is independent of time, so we must have $$\nabla \times (\nabla \times \mathbf{b}) = 0$$ and $\nabla \times \mathbf{E}_0 = -\mathbf{b}$ with $\nabla \cdot \mathbf{E} = \nabla \cdot \mathbf{E}_0 = 0$.

This can admit non-zero solutions for $\nabla \times \mathbf{b}$ so that $\mathbf{E}$ need not be independent of time.

 

[Shreya]

@pasmith, thank you so much for the answer. I understood your reply except the last statement.

This can admit non-zero solutions for ∇×b so that E need not be independent of time.

Could you please explain this?
Could you explain how a changing magnetic field, creates a change electric field and that creates a changing magnetic field and so one. I thought that every time you take a derivative the power of time decreases until you get a zero.
P.s I am just a highschooler so I am not very comfortable with multivariable calculus, but I can quite understand what you wrote above. Could you please elaborate?

 

[TSny]

Homework Statement: I know that a time varying magnetic field, creates a curling electric field around it. My question is this: If the time variance is linear, the curl of electric field is constant. Doesn't that imply electric field is not varying with time? So, no curling magnetic field is generated?

For a specific example, consider the $E$ and $B$ fields produced by charging parallel, circular plates:

The lower plate is being charged positive while the upper plate is being charged negative. Neglecting edge effects, the E field is parallel to the z-axis and is increasing in strength while the $B$ field lines are circular. We assume that the rate of charging is controlled such that the magnitude of $B$ at any point increases linearly with time. Then the $E$ field turns out to be nonuniform in space and increases quadratically in time. The magnitudes of the fields may be expressed as $$E = at^2 + \frac{ar^2}{2c^2}$$ $$B = \frac{art}{c^2} .$$ Here, $a$ is a constant with dimensions of electric field divided by time squared.
$r$ is the horizontal distance of a point from the z-axis.

It can be checked that these fields satisfy all 4 of Maxwell’s equations. $B$ increases linearly with time. $E$ contains a time-dependent term (quadratic in $t$) which is uniform in space and a time-independent term that is nonuniform in space (quadratic in $r$). These two terms correspond to the right-hand side of @pasmith 's

$$\mathbf{E} = \frac{c^2t^2}{2} (\nabla \times \mathbf{b}) + \mathbf{E}_0(\mathbf{x})$$

In the figure, you can see how the $B$ field “curls” around the changing E field.

Even though the $E$ field lines are straight, the $E$ field nevertheless has a nonzero curl (in the mathematical sense) around the changing $B$ field. This is due to the increase in $E$ as $r$ increases.

 

[Shreya]

Wow! Thank you so much, @TSny and @pasmith. I got it, now! I love electromagnetism and this awesome community