What is the direction of the induced electric field when magnetic field moves towards wire loop?

Question

Follow up to my previous question,

I have doubts about the direction of the induced electric field on the stationary wire loop

I thought that since the magnetic field is increasing out the page, $\nabla \times \vec{E}$ must point into the page, so electric field will flow in the clockwise direction, following current in the loop.

I then realized that my initial reasoning could be wrong. If we remove the wire loop from the picture and focus on how the magnetic field changes with respect to time, $\frac{\partial \vec{B}}{\partial t}$ will point leftwards (-x direction). That means $\nabla \times \vec{E}$ must be pointing in +x direction and the induced electric field must point downwards, as shown in the rough 3D sketch below

Here, the black rectangle is the stationary wire loop and the red rectangle is the moving magnetic field source. The orange arrow represents $\frac{\partial \vec{B}}{\partial t}$ and the blue arrows represent electric field lines around the wire loop. I am not sure whether the blue arrow 1 or blue arrow 2 is responsible for creating current in the wire loop but since Faraday's law tells me that the induced current should flow clockwise so the blue arrow 1 must be responsible but I am not sure what role the blue arrow 2 plays here.

Answer 1

$\frac{\partial\boldsymbol{B}}{\partial t}$ is not to the left. If it were, the x-component of the magnetic field would grow and the B-field vectors would appear to tip to the left until they were essentially in the plane of the wire hoop.

Faraday's law may also be cast in integral form:

$-\frac{\partial\Phi_A}{\partial t} = \oint_{\partial A} \boldsymbol{E}\cdot\boldsymbol{dl}$

where $\Phi_A$ is the flux of the magnetic field through some area and $\partial A$ is the closed loop boundary around that area.

For this problem let's take "out of the page" to be the positive direction. Then the wire loop is a boundary enclosing a particular area through which the magnetic flux is increasing as the loop enters the magnetic field region. So Faraday's law says the loop integral of electric field around the wire should be negative, or into the page. This results in a clockwise curling electric field which generates the current.