Electricity (6/11) – Electric Field, Magnetic Field, Current, and Voltage (EBIV) Relationships

Transformer EMF: Moving Magnetic Field and a Stationary Conducting Coil

In the last section we moved a coil over a stationary magnetic field and found that it is the v̅ x B̅ component of the Lorentz force that creates the EMF that causes the current to flow. 

We want to imagine now that we are subjecting a stationary conducting coil to an changing magnetic field.

Picture: Moving Magnetic Field and Stationary Conducting Coil

Let’s flip the picture so we have a plan view of the coil (looking down on it from above).

This is the view you see below where

• the solid black circle is the conducting coil and
• X represents the magnetic field going into the page (symbolized as the fletching, the back end, of an arrow).

Picture: Planar View: Changing Magnetic Filed with Fixed Conducting Coil

As B_o  , the original or external magnetic field, changes, a current I will begin to “flow” in the coil.

What is the responsible force that makes this happen?  Let’s start with our EMF general equation:

EMF=∮(E̅ + v̅ x B̅)⋅dL̄ ; EMF Expression Using the Lorentz Force Components

There are no moving electrons to begin with in this example, so the v̅ x B̅ term is zero!

EMF= ∮Ē•dL̄ ; Transformer EMF;  Moving Magnetic Field  and Stationary Conducting Coil

So it’s an Electric field E that forces the electrons around the coil.

• But, very importantly,  this E field is NOT an electrostatic field (which is a conservative field)
• E is a very unique , looping, non-conservative Electric field that is able to push the electrons around as current. 

More on this E field later , but now lets go through the sequence of events that occur after this E field is created (use the picture above as you read). 

• Bo changes.
• A non conservative , looping, electric field E_ind is induced.
• E_ind forms concentric closed loop circles that move counterclockwise.
• This induces a current I which
• induces a magnetic field B_ind
  • We know B_ind will be opposite B_o  according to Lenz’s Law.
  • This is represented as the brown dots in the drawing above indicating the induced B field is coming out of the page.
  • We can then use the right hand rule (Right hand thumb towards I and curled fingers towards B) to determine the direction of I which will be counter-clockwise.

Let’s come back to the E_ind .

E_ind is not the same as the electrostatic E field, E_{electrostatic}

So, in a changing magnetic field and a stationary conducting coil, an induced electric field is created and applies the force to move the current.

But E_ind  is distinctly different than the electrostatic field E_{electrostatic} .

• E_ind  makes loops (like B fields); has no explicit charge source; not static and always changing
• E_{electrostatic}  is created by stationary static charges ; i.e. it starts and stops on a charge and is static

E_{electrostatic}  field lines  extend radially outward or inward. These fields are conservative, which means the work done by the field on a charged particle moving in a closed loop is always zero.

• So we can say the closed loop line integral on the circuit (i.e. work done) is zero i.e.  W/q =  ∮Ē•dL̄ = 0
• This is because the force exerted by the field is independent of the path taken and
• only depends on the starting and ending points.
• This property allows us to define an electric potential, where the work done is simply the change in potential energy between two points.

E_ind , generated by a changing magnetic field, are non-conservative.

• The work done by  E_ind on a charged particle moving in a closed loop is not zero.
• There will therefore be no well defined electric potential.

• The EMF term, then, is W/q =  ∮Ē_ind•dL̄ 

So we can write Faraday’s Law as follows:

∮Ē_ind•dL̄ = −NdΦ_B/dt  ; Transformer EMF ; Faraday’s Law of Electromagnetic Induction (line integral form relating E to B)

• This equation describes the phenomenon known as Transformer EMF (or non-motional $\text{EMF}$), where the electric field is induced by a time-varying magnetic field.

• Name: It has different names
  • The Maxwell-Faraday Equation (Integral Form) – One of the four Maxwell Equations.
  • Faraday’s Law of Induction – Most common and general name
• Physical Meaning: The line integral of the induced electric field (E_ind) around a stationary closed loop is equal to the negative rate of change of the magnetic flux (Φ_B) through the surface bounded by that loop.
• Requirement: The loop of integration (dL̄) must be stationary (fixed in space).

• E_ind is the electric field vector. It is an induced non conservative field (its not an electrostatic field).

• dL̄ is the infinitesimal vector of a path element along a closed loop.
  • The direction of this vector is tangent to the path.
• ∮ is a closed line integral. It means you are summing up the dot product of Ē and dL̄ along an entire closed loop.
  • • The result of this integral is the electromotive force (EMF), which is the work done per unit charge in moving a charge around the closed loop.
    • For induced electric fields, this value is non-zero, indicating that the field is non-conservative.

• N: # of coils 

• ΦB is the magnetic flux, which is a measure of the total number of magnetic field lines passing through a given surface
• dΦ_B/dt = Is the rate of change of magnetic flux with time
• −: The negative sign represents Lenz’s Law.
  • It indicates that the direction of the induced electric field (and thus the induced current) will be such that it creates a magnetic field that opposes the change in the original magnetic flux.
  • This is a fundamental statement of energy conservation in electromagnetism.

We also know that 

Φ_B  = ∫_sB̄•dĀ so let’s substitute this into the expression above to get

∮Ē_ind•dL̄ = −NdΦ_B/dt = -Nd/dt∫_sB̄•dĀ ; Transformer EMF ; Faraday’s Law of Electromagnetic Induction (line integral form relating E to B)

The Differential Form (The Maxwell-Faraday Equation)

This is the most fundamental and generalized form, fitting in as the final piece.

It extends the concept from a physical loop to any point in space, showing that a time-varying magnetic field (B) creates a circulating electric field (E) at a fundamental level, even in a vacuum.

It is one of the four Maxwell’s Equations and is mathematically derived from the integral form using the Divergence Theorem and  Stokes’s Theorem.

• See my article, Maxwell Equations: From Integral to Differential Forms, for the derivation. 

∇×Ē=-∂B̄/∂t ; Faraday Equation (Differential Form)

This equation shows how fields themselves interact dynamically.

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