# A question regarding Faraday's integral law

This might be a quickly phrased question so I apologize if I've missed something trivial in my understanding.

Faraday introduced the concept of fields to explain the phenomenon of action at a distance (E.g: It is believed that Newton wasn't satisfied with his law of gravitation because there's nothing in between the Sun and Earth to "carry" the forces). Faraday proposed that fields physically exist and they are the mediators that transfer forces.

The scenarios 1 and 2 (ref. figure) can be explained by Faraday's integral law : $$\oint_C \vec{E}\cdot d\vec{r} = \frac{d(\int_S \vec{B}\cdot d\vec{A})}{dt}$$

Consider scenario 1:

A loop of wire moves with a velocity v towards a stationary magnet. The magnetic field lines pass through the wire (not just the interior of the loop, but also through the charges present in the wire). An explanation for the generation of electric current in the wire as a result, comes from the application of Lorentz force equation $$\vec{F}=q (\vec{v} \times \vec{B})$$. Since the charges are moving in this frame of reference through magnetic field lines, a force is generated that makes them move and generates the current. This is very much in line with what Faraday wanted. The magnetic field explains the action at a distance.

But consider scenario 2 :

We know that the magnetic field generated by the toroidal coil is only present inside the cross-section of the toroid and zero outside. Now if you pass an external wire through the toroid on the inside, close it from the outside with a galvanometer and switch on the current passing through the toroidal coil, the galvanometer needle will get deflected. Even though there are no magnetic field lines passing through the physical wire (Not the interior of the region formed by the closed loop of wire) since the magnetic field outside the toroid is zero, electric field is still generated in the loop.

Shouldn't this experiment (scenario 2) have been a setback for Faraday? Because he tried to explain action at a distance by showing that it is the field lines that physically mediate the forces between two objects and scenario 2 still falls under "action at a distance". The coil "knows" that the magnetic field inside the area enclosed by the loop of coil is changing and charges start moving, without any magnetic field lines physically going and "talking" to the charges on the wire. Did he try to resolve this question?

I think my doubt can be summarized in the following question : Since electric field is created in the loop that moves the charges, shouldn't Faraday have elaborated on how changing magnetic field lines in one place create electric field lines in another place without any contact. Also, according to Faraday, Does the circulation of electric field around a closed contour instantaneously respond to the rate of change of magnetic flux through a surface bounded by the contour? It can't be, according to "No information can travel faster than light".

I read only the last paragraph. Faraday's law simply does not imply that changing magnetic field lines in one place (no need to say "lines," but perhaps helpful for intuition) instantaneously create changing electric field lines in another place. The Maxwell-Faraday law is

$$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}.$$

If we are working in $$(x,y,z)$$ space, and time is $$t$$, then all vector fields are functions of $$(t,x,y,z)$$. Then the law is, more specifically,

$$(\nabla \times \mathbf{E})(t,x,y,z) = - \frac{\partial \mathbf{B}}{\partial t}(t,x,y,z).$$

That's all. No information travels instantaneously. The curl of one vector field at a specific point in space and a specific time is equal to the negative time derivative of another at the same point and at the same time.

Now, sure, we may use Stokes Law to integrate both vector fields along convenient contours/ surfaces to obtain things like

$$\text{emf} = - \frac{d\Phi_B(t)}{dt} = -\int_{\Sigma} \mathbf{B}(t)\cdot d\mathbf{A}$$

but this is no longer a statement about vector field lines, it's a statement about derived quantities like emf and flux. Does this help?

• I understand your point. As you say, the integral form of Faraday's law (after applying Stoke's law ; with time coordinate written explicitly) is : $\oint_C \vec{E}(t,\vec{r})\cdot d\vec{r} = \frac{d(\int_S \vec{B}(t,\vec{r})\cdot d\vec{A})}{dt}$. Could you now refer to scenario 2 in the question above. Doesn't this seem to imply an instantaneous generation of electric field lines that circulate around the toroid, as soon as the switch is closed? – Ajay Mohan Mar 31 '19 at 10:28
• Just to answer my own question in the comment above, for the sake of completeness in case anyone stumbles upon this question. The mistake here is to use magnetostatics (Ampere's law, Biot Savart law) to determine the magnetic field generated by the toroid when the current isn't static but changing. This is called the Quasistatic approximation. I overstepped the limits of this approximation when I used it in this problem and found the approximate solution. In the true solution, electromagnetic news still travels at the speed of light. – Ajay Mohan Sep 14 '19 at 4:38