What is the current induced in a straight wire when a magnetic field is turned from on to off?

Consider a straight wire which runs along the y-axis. A B-field exists which is uniform and only in the x-direction:

$$\vec{B}=(B(t),0,0)$$

The B-field at time $t_0$ is 'switched off' in a sharp step, which will be modelled by the tanh function:

$$B(t)= \begin{cases} B_0 & t\leq t_0 \\ 0 & t>t_0 \end{cases} \approx \frac{B_0}{2}\left [ \tanh(100(t-t_0)) +1 \right ]$$

What is the current induced in the wire due to the change in magnetic field as a function of time?

What I have tried is to work out the E-Field y-component using Maxwell's 3rd law (using the notation $\partial_{yz} x \equiv \frac{\partial^2 x}{\partial y \partial z}$):

$$\nabla\times\vec{E} = -\partial_t \vec{B}$$

But the B-field only has an x-component, so:

\begin{align} \partial_y E_z - \partial_z E_y &= -\partial_t B(t)\tag{i} \\ \partial_z E_x - \partial_x E_z &= 0\tag{ii} \\ \partial_x E_y - \partial_y E_x &= 0\tag{iii} \end{align}

So if you differentiate (i) by x, (ii) by y and (iii) by z:

\begin{align} \partial_{xy} E_z - \partial_{xz} E_y &= -\partial_{xt} B(t) \\ \partial_{yz} E_x - \partial_{xy} E_z &= 0 \\ \partial_{xz} E_y - \partial_{yz} E_x &= 0 \end{align}

But then if you sub them all in together you get:

$$\partial_{xt} B(t) = 0$$

Which isn't very useful.

• If I want to work out the current induced in the wire which runs along the y-axis do I want to work out: $E_y$? And If so how do I do so?
• Would current be induced in any other direction?
• What did you try? – Bernhard Mar 16 '13 at 23:58
• @Bernhard I have included what I have tried now – joshlk Mar 17 '13 at 0:37
• What is the derivative of $B$ at time $t_0$ ? – DilithiumMatrix Mar 17 '13 at 0:48
• @zhermes Either infinite or not defined I suppose. But my deviation hasn't used any information about B yet – joshlk Mar 17 '13 at 0:58
• Hi Josh - this is a site for conceptual questions, which means you should try to ask something more specific than "Any help would be much appreciated." What concept is confusing you about this problem? Is there something you think you might be able to use, but you're not sure if you can? That sort of thing... if you can make an edit like that, I'll be very happy to reopen this. Just include @David in a reply comment to get my attention. – David Z Mar 17 '13 at 1:58