User:Metacomet/Emf

Time-varying magnetic field
A time-varying magnetic field is any magnetic field which has an associated magnetic flux density B such that


 * $$ {\partial \mathbf{B} \over \partial t} \ne 0 $$

A non-time-varying magnetic field (also known as a constant-in-time magnetic field) is one for which


 * $$ {\partial \mathbf{B} \over \partial t} = 0 $$

Slowly-varying magnetic field
A slowly-varying magnetic field is one where the magnetic flux density varies only slowly with respect to time. More specifically, the time rate of change of the magnetic flux density B must meet the following condition:


 * $$ \bigg| {\partial \mathbf{B} \over \partial t} \bigg| << \bigg| { \mathbf{B} \over \tau } \bigg| $$

where
 * $$ \tau = {d \over c} $$
 * d is the maximum distance that a signal would have to propagate in the physical structure, and
 * c is the speed of light in the medium.

Assume the magnetic flux density takes a time-harmonic form with angular frequency &omega;, then these restrictions are equivalent to saying:


 * $$ \omega << {1 \over \tau} $$

or
 * $$d << { \lambda \over 2 \pi } $$

where


 * $$ \lambda = { 2 \pi c \over \omega } $$ is the wavelength.

Electro-quasi-static approximation
In the Electro-quasi-static approximation, or EQS, we assume that the only magnetic fields are ones that are varying slowly with time, as defined above.

Formal definitions of emf and electric potential
A couple of things:

1. The line integral of a vector field is by definition a scalar. The issue is that you cannot define a unique scalar field at all points in space because the line integral of the electric field is not in general path independent. But the integral itself is definitely a scalar. It's just not a unique function of space.

2. I think we need to agree on the definitions of both emf and electric potential before we proceed.

3. According to the reference from which you quoted, the definition of emf is:


 * $$ \mathcal{E} = \oint_C \mathbf{E} \cdot d\mathbf{l} $$

Furthermore, this definition is valid under all circumstances, regardless of whether there is a time-varying magnetic field or not.


 * Comment: this definition cannot be right.  If it is, that means that the emf in most electrical circuits, in the absence of a time-varying magnetic field, is zero.


 * This is only the emf due to varying mag fields, and fails to take account of the electrostatic e.m.f. from batteries etc! In my book, it implies that e.m.f is only created by induction. In the case of static fields, emf as such is not really mentioned.


 * Here any voltage difference is given by
 * $$ \mathcal{V}_a^b = -\int_a^b\mathbf{E} \cdot d\mathbf{l} $$

That's fine, as long as you understand that a voltage difference, even in an electrical circuit, is identical to a potential difference between the two points in space which coincide with the two nodes in the circuit. In other words,

$$ \mathcal{V}_a^b = \mathcal{V}_b - \mathcal{V}_a = \Phi(b) - \Phi(a) = \Phi_a^b $$

In fact, that is arguably a pretty good definition for the voltage drop between two nodes in a circuit. Again, just because we are talking about an electric circuit doesn't mean we cannot use Maxwell's Equations. In fact, Maxwell's Equations are completely general, and apply in any situation. Electrical circuis are simply a special case.

4. Electric potential, on the other hand, is defined if and only if there is no time-varying magnetic field passing through the surface enclosed by Contour C, or if the time rate of change of the magnetic field is small enough so that its impact is insignificant. Under these limited conditions, we can say that the curl of the electric field is zero:


 * $$ \nabla \times \mathbf{E} = 0 \qquad \qquad {\partial \mathbf{B} \over \partial t} = 0$$
 * OR
 * $$ \nabla \times \mathbf{E} \approx 0 \qquad \qquad \bigg| {\partial \mathbf{B} \over \partial t} \bigg| << \bigg| { \mathbf{B} \over \tau } \bigg| $$


 * where $$ \tau = {d \over c} $$ and d is the maximum distance that a signal would have to propagate in the physical structure. These restrictions are equivalent to saying:


 * $$ \omega << {1 \over \tau} $$  or $$d << { \lambda \over 2 \pi } $$


 * Under these conditions, and only under these conditions, we can define a unique scalar function, called the electric potential, such that the electric field is the negative of the gradient of the electric potential:


 * $$\mathbf{E} = - \nabla \Phi $$

5. Using vector calculus, the last equation is equivalent to:


 * $$ \Phi(b) - \Phi(a) = - \int_a^b \mathbf{E} \cdot d\mathbf{l} $$


 * for any two points a and b. But again, this electric potential difference is defined if and only if there is no time-varying magnetic field passing through the region of space containing points a and b.

6. This last equation also provides the basis for Kirchoff's voltage law:


 * $$\oint_C \mathbf{E} \cdot d \mathbf{l} = 0 $$


 * with the condition, again, that there are no time-varying magnetic fields passing through the surface enclosed by Contour C, or that any such magnetic field is varying slowly enough that its effect is insignificant.

A more general approach
From Gauss's law for magnetism, we have:


 * $$ \nabla \cdot \mathbf{B} = 0 $$

From vector calculus, there is an identity which states:


 * The divergence of the curl of any vector field A is always zero:
 * $$\nabla \cdot ( \nabla \times \mathbf{A} ) = 0 $$

Thus, we can define a vector field that is uniquely defined at all points in space such that the curl of this so-called vector potential A is equal to the magnetic flux density B:


 * $$ \mathbf{B} = \nabla \times \mathbf{A}$$

or equivalently,


 * $$\int_S \mathbf{B} \cdot d \mathbf{a} =   \oint_C \mathbf{A} \cdot d \mathbf{l} $$

From Faraday's law, we have:


 * $$\nabla \times \mathbf{E} = - {\partial \mathbf{B} \over \partial t}$$

Combining the last equation with the differential form of the prior equation, and simplifying, we have:


 * $$\mathbf{E} = - \nabla \Phi - {\partial \mathbf{A} \over \partial t}$$

where &Phi; is the electric potential, as before, except that we are now allowing time-varying magnetic fields in this region.

If we now take the definition of emf, as cited above, we have
 * $$ \mathcal{E} = \oint_C \mathbf{E} \cdot d\mathbf{l} = - \oint_C { \partial \mathbf{A} \over \partial t } \cdot d\mathbf{l} = - {d \over dt } \int_S \mathbf{B} \cdot d \mathbf{a} = - \, { d \Phi_B \over dt } $$


 * Comment: So here again, we have a problem, because this last equation suggests that the emf is always zero in the absence of a time-varying magnetic field.

I believe that a better and more general definition of emf would be:


 * $$ \mathcal{E}(b) - \mathcal{E}(a) = \int_a^b \mathbf{E} \cdot d\mathbf{l} $$