How is electromotive force (EMF) induced? (AP Physics II Class Material)

 
Case I. Conductor loop entering a magnetic field
  We've learned that induced electromotive force (EMF) is induced by changes in magnetic flux inside a closed loop, but this is actually nothing new that is naturally derived by known physics. Either magnetic or electric force can explain these induced EMF. We will explore these facts, as well as a new insight from relativity.
  First, consider the situation (Figure 1) where a straight wire is pulled with velocity $\vec{v}$ for a uniform magnetic field $\vec{B}$ entering perpendicularly to the surface, resulting in an increasing magnetic flux.

Fig 1

In this situation, let's try to calculate inner product of the magnetic force $\vec{v} \times \vec{B}$ and the all line element $\vec{dl}$ along the closed loop $\partial A$ ($\partial$ means "boundary" and $A$ means "area"). This is not exactly a "work" done by the magnetic force, as it is not always doing work.
\begin{equation}
    \int_{\partial A} (\vec{u} \times \vec{B}) \cdot \vec{dl} = \int_{\partial A} (\vec{dl} \times \vec{u}) \cdot \vec{B} = \int_{\partial A} (\vec{dl} \times \vec{v}) \cdot \vec{B} = -\int_{A} \vec{B} \cdot \frac{\vec{da}}{dt} = -\frac{d}{dt} \Phi = \epsilon.
\end{equation}
In the last part, the area changes with time, but the magnetic field does not, so it was possible to do a total differentiation of $\vec{B} \cdot \vec{a}$. From this result, we can speculate that the induced EMF is caused by the magnetic force acting along the boundary of a closed conducting loop.

Case II. Conductor loop entering a magnetic field
  Just as the movement of a charge induces a rotating magnetic field,
\begin{equation}
    \oint \vec{B} \cdot \vec{dl} = \mu_0 I_{\text{in}},
\end{equation}
according to Ampere-Maxwell's law, a changing magnetic flux induces a rotating electric field similarly:
\begin{equation}\label{eq1}
    \oint \vec{E} \cdot \vec{dl} = -\frac{d\Phi}{dt}_\text{in} \text{ or equivalently } \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}.
\end{equation} 
  Figure 2 illustrates the similar nature of magnetic field and induced electric field. As the source of the magnetic field was a current, by analogy, \textit{a charge flux}, the source of the induced electric field is a \textit{change in magnetic flux}. However, since their signs are reversed, the direction of the magnetic and induced electric fields for a source in the same direction should be opposite.

Fig 2

  We now consider the case where the wire is stationary and the magnetic field is moving. (Figure 3) 

Fig 3

The change of the magnetic flux is equivalent to an infinite number of straight wires in a row, analogous to an electric current, creating an induced electric field as shown in the Figure 3. The change of magnetic flux in unit time is
\begin{equation}
    \frac{d\Phi}{dt} = \frac{d}{dt} \left\{ \vec{B} \cdot \left( \int \vec{dl} \times \vec{v} \right) dt \right\}= -Blv,
\end{equation}
which is, by (\ref{eq1}), equal to the electric force exerted on a unit charge by this induced electric field
\begin{equation}
    \oint_{\partial A} \vec{E} \cdot \vec{dl} = 2El = Blv.
\end{equation}
Therefore, in this case, the source of the EMF is nothing but the power (work per time) of the charge inside the wire subjected to the electric field $E=Bv/2$.


Case I vs II. Implications
  Comparing Case I and II, there are some implications we should note. 
  First, if the principle of relativity is true, then Case I and II should be "physically equivalent" situations. However, in Case I, the induced EMF is magnetic, and in Case II, it is electric. What does this mean? Depending on the coordinate system in which we observe, it can be either an electric or magnetic field, meaning that electric and magnetic phenomena can actually be lumped into the same category. This is why we conventionally called "electromagnetic" field. It is roughly ture that a varying electric field induces a magnetic field and vice versa. Instead of seeing them as a picture of a relationship in which one induces the other, why don't we start today by seeing them as a single entity: "electromagnetic" field?
  Second, there is no place for the concept of "particles" in electromagnetic phenomena. We have assumed a particle inside a wire of charge $q$, but in reality, the wire only acts as a constraint on the area in which an arbitrary particle can move around, and the essence of the induced electromotive force is in the electromagnetic field itself. In fact, electromagnetic theory is a pure "field theory," and all electromagnetic phenomena can be described in terms of "fields" without "particles." In modern physics, the concept of a particle is also interpreted as a "quantum excitation of a field." In effect, the particle mechanics we've learned about is merely an approximation of field theory, and the true nature of physics resides in the fields.

Exercise
  You have a circuit with two possible connections to switches $a$ and $b$, as shown in the figure below. A magnetic field $\vec{B}$ entering perpendicularly is placed on the circuit. If you first connect to $a$ and then immediately switch to B, is EMF induced? If so, calculate EMF. If not, explain why not. (When connected to switch $a$, the area is $A$, and when connected to $b$, the area is $2A$.)

Fig 4