Induced EMF in single stationary wire Suppose we have a conducting stationary wire in a uniform magnetic field: $$\mathbf B(t) = kt \mathbf u_z$$ with $k>0$. Assume the wire is a segment that lies on the $xy$ plane and its length is arbitrary (I will assume it equal to $L$). We want to know the induced electromotive force ($\mathcal E$) on the wire.
I know from Maxwell equations that:
$$\nabla \times \mathbf E = - \frac{\partial \mathbf B}{\partial t}$$
Which implies that $\mathbf E$ has curl equal to $\mathbf A = -k\mathbf u_z$. Finding $\mathbf E$ is impossible (as far as I know) because the vector potential of $\mathbf A$ isn't unique.
I have no idea how to proceed and any help would be greatly appreciated.
I have seen this similar question but the wire I am considering is stationary. I have also seen this other question but it doesn't explain how to find $\mathcal E$.
 A: Given the homework-ish nature of this problem, I will only provide a useful hint. First off, since you already have a physical $B$-field there is no need to fix a gauge, since, like I said you already have the physical quantity.
Now, as you stated what you therefore know is that 
$$\nabla \times \mathbf{E} = -k   \hat{z}$$
(If you assume that the B-field has always been on).
Hint: Multiply both sides by $d\mathbf{A}$ and integrate. This means that 
$$\iint \nabla \times \mathbf{E} \cdot d\mathbf{A} = -k  \iint \hat{z} \cdot d\mathbf{A} $$
Now, choose $dA \hat{z}$, and use stokes theorem so that we may write 
$$ \oint_{\partial A} \mathbf{E} \cdot d\mathbf{l} = -k  \iint_A \hat{z} \cdot d\mathbf{A} $$
where $A$ is an imaginary loop of area $A$ enclosing the wire whose orientation is defined by the normal vector $\hat{z}$. This is a simple circuital law, and I trust you can take it from here.
Hint 2: Since the wire length is arbitrary, take it to be very large. Recall, Faraday's law of induction 
$$ \mathcal{E}= \oint_{\partial A} \mathbf{E} \cdot d\mathbf{l}. $$
A: I wonder if the problem is not ill posed ?
The electric field associated with the magnetic field depends on the details of the circuit that generates it. Is the magnetic field generated by a solenoid of circular section or square section ? .... Is it centered on the axis?
The details of the electric field have no importance for a closed circuit but for an open circuit they are important !
Imagine that the magnetic field is created by a solenoid of circular section centered on the axis. The cylindrical symmetry easily gives the electric field from the Maxwell faraday equation : $\overrightarrow{E}=-\frac{k}{2}r\overrightarrow{{{e}_{\theta }}}$ 
For this orthoradial electric field, the circulation of the field depends on the position of the segment in the plane Oxy. If the segment is radial, it is zero. If it is normal to the radius, it may be different from $0$ ...
So ill posed problem ?
