Classical physics is our model of the laws that govern the
behavior and interactions of macroscopic objects in the world around us.
All the laws of classical physics were known by the end of the 19^{th}
century. Classical physics works well describing and predicting almost all
everyday phenomena, but it starts to fail when things get too ... .
(You can insert any extremes here, for example big, small,
hot, cold, fast, etc.)

In classical physics the world is made of matter
**particles** that behave according to Newton's
laws of motion. To use Newton's laws, we need to know which forces are acting
on the particles. The interactions that give rise to the forces between the
particles are represented by **fields**.
Electric and magnetic fields represent the electromagnetic interactions. If we
know the fields, we know the electric and the magnetic forces acting on charged
particles.

**F** = **F**_{electric} + **F**_{magnetic} = q**E** + q**v** × **B** = q(**E** +
**v** × **B**).

In classical physics our model for the electromagnetic
fields is a set of four equations, called **Maxwell's
equations**. They let us predict **E** and
**B**. They have a broader range of applicability than Newton's laws. They are relativistically
correct and correctly describe the fields when relative speeds approach the
speed of light. But the classical model for the electromagnetic fields no
longer describes observations correctly on the scale of atoms or elementary
particles.

Maxwell's equations predict that **static
electric fields**, i.e. field that do not change with time, are produced by charges. Charges are
sources and sinks of field lines. Static electric field lines begin on positive
and end on negative charges.

Static electric field
lines **NEVER** form loops.

The static electric field is a
conservative field. If you do work on a charge against the static electric
field, this work is stored as potential energy. The potential energy is a
function of position only. You can let the electric field do work to
recover the energy you expended in moving the charge.

In statics, Maxwell's equations for the electric field
combine to give you Coulomb's law and the principle of superposition.
Everything else is just vector addition.

Maxwell's equations predict that **static magnetic** fields are
produced by moving charges or currents. There are no magnetic charges, and
therefore no sources and sinks for magnetic field lines.

Static magnetic fields lines
**ALWAYS** form closed loops.

They encircle moving charges according to the right hand rule.

If we do NOT restrict ourselves to steady state conditions, two additional terms appear in Maxwell's equations.

(1) | ∮_{A} E∙dA = Q_{inside}/ε_{0} |

(2) | ∮_{Γ} E∙ds
= -∂/∂t∫_{A}B·dA. |

(3) | ∮_{A} B∙dA = 0 |

(4) | ∮_{Γ} B∙ds = μ_{0}I_{through Γ}
+ (1/c^{2})∂/∂t∫_{A}E·dA. |

Maxwell's equations tell us that there is another source of electric and magnetic fields, when either of those fields changes with time.

- Changing magnetic fields are a source of electric fields.
- Changing electric fields are a source of magnetic fields.

The key word is CHANGE. The fields produced by this CHANGE are called dynamic or **induced fields**. The induced fields do not
have sources or sinks.

Field lines of induced, dynamic fields **ALWAYS** form closed loops. They always have **circulation**.

Mathematical definition of circulation:Take any closed loop, for example a circle. At a fixed
time, move in small steps ∆

The circulation of **E** around a closed loop = lim_{∆r-->0}Σ_{around loop}
**E∙**∆**r** = ∫_{around loop}**E·**d**r**.

For induced fields, the circulation is always non-zero.

Maxwell's equations tell us about the magnitude and the direction of the circulation. One of the equations is called
**Faraday's law**. For the absolute value
or magnitude, Faraday's law states that the circulation of the
electric field **E** around any closed loop is equal to the rate of change
of the magnetic flux through the area enclosed by the loop.

The flux of a vector field through a surface area is the amount of whatever the field represents passing through the area. The total flux depends on strength of the field, the size of the surface area it passes through, and on how the area is oriented with respect to the field.

We have defined the **electric flux** as dΦ_{E }= E dA cosθ, Φ_{E }
= ∫_{A}**E**·d**A**.

The **magnetic flux** is defined in an
analogous way as dΦ_{B }= B dA cosθ, Φ_{B} = ∫_{A}**B**·d**A**.

Another of Maxwell's equations is the **Ampere-Maxwell
law**. It states that the circulation of the magnetic field **B**
around any closed loop is the sum of two terms. The static term is proportional
to the current flowing through the loop and the dynamic term is proportional to
the rate of change of the electric flux through the area enclosed by the loop.

This interplay between induced electric and magnetic fields leads to the
production of electromagnetic wave.

**Electromagnetic waves** are solutions to Maxwell's equations, even in free space, when
no charges or currents are present.

Find the magnetic flux through each of the coils below.

(a) A magnetic field of magnitude B = 5 mT points along the normal of a circular turn of wire of radius r = 1 cm.

(b)
A magnetic field of magnitude B = 5 mT makes an
angle θ = 20^{o} with the normal of triangular coil of 50 turns
with base b = 5.0 cm and height h = 4 cm.

Solution:

(a) Φ_{B }= B A cosθ = B A = (5*10^{-3} T)(π*10^{-4}
m^{2}) = 1.57*10^{-6} Tm^{2}.

(b) Φ_{B
}= B A cosθ. Since the coil has 50
turns, A = 50 times the area of the triangle.

Φ_{B }= (5*10^{-3} T)(50*5*10^{-2}*2*10^{-2}
m^{2})*cos(20^{o}) = 2.35*10^{-4} Tm^{2}.

A rectangular loop of N turns is rotating about an axis in a region of uniform magnetic field, as shown in the figure on the right.

Describe how the magnetic flux through the loop is changing with time.

Solution:

Φ_{B} = B A cosθ = B N A_{rectangle} cosθ.

The product B N A_{rectangle} is constant, but θ, the angle
**B** makes with the normal, is changing with time.

(For a flat area you can choose the direction of the normal. Assume a
flat area lies in the xy-plane. The normal to this area is the
z-axis, for its direction you can choose the positive or negative
z-direction.)

If the magnetic field at t = 0 is aligned with
the normal, then the flux has its maximum positive value at t = 0.

It deceases towards zero as θ approaches 90^{o}, reaches its
maximum negative value at θ = 180^{o}, approaches zero again
as θ approaches 270^{o}, and reaches its maximum positive
value again at θ = 360^{o}.

The flux is continuously
changing.