Electric and Magnetic Constants

In the equations describing electric and magnetic fields and their propagation, three constants are normally used. One is the speed of light c, and the other two are the electric permittivity of free space ε0 and the magnetic permeability of free space, μ0. The magnetic permeability of free space is taken to have the exact value

This contains the force unit N for Newton and the unit A is the Ampere, the unit of electric current.

Electric and Magnetic Constants

With the magnetic permeability established, the electric permittivity takes the value given by the relationship

where the speed of light c is given by

This gives a value of free space permittivity

which in practice is often used in the form

Physical Connections to Electric Permittivity and

Magnetic Permeability

Expressions for the electric and magnetic fields in free space contain the electric permittivity ε0 and magnetic permeability μ0 of free space. As indicated in the section on electric and magnetic constants, these two quantities are not independent but are related to "c", the speed of light and other electromagnetic waves.

Physical Connections to ε0 and μ0

The electric permittivity is connected to the energy stored

in an electric field. It is involved in the expression for

capacitance because it affects the amount of charge

which must be placed on a capacitor to achieve a certain

net electric field. In the presence of a polarizable medium,

it takes more charge to achieve a given net electric

field and the effect of the medium is often stated in terms

of a relative permittivity.

The magnetic permeability is connected to the energy stored in a magnetic field. It is involved in the expression for inductance because in the presence of a magnetizable medium, a larger amount of energy will be stored in the magnetic field for a given current through the coil. The effect of the medium is often stated in terms of a relative permeability.

Electric Field Energy in Capacitor

The energy stored on a capacitor is in the form of energy density in an electric field is given by:

This can be shown to be consistent with the energy stored in a charged parallel plate capacitor

Energy in Capacitor

When the switch is closed to connect the battery to the capacitor, there is zero voltage across the capacitor since it has no charge buildup. The voltage on the capacitor is proportional to the charge

Storing energy on the capacitor involves doing work to transport charge from one plate of the capacitor to the other against the electrical forces. As the charge builds up in the charging process, each successive element of charge dq requires more work to force it onto the positive plate. Summing these continuously changing quantities requires an integral.

Capacitor Energy Integral

Transporting differential charge dq to the plate of the capacitor requires work

But as the voltage rises toward the battery voltage in the process of storing energy, each successive dq requires more work. Summing all these amounts of work until the total charge is reached is an infinite sum, the type of task an integral is essential for. The form of the integral shown above is a polynomial integral and is a good example of the power of integration.

Energy in an Inductor

When a electric current is flowing in an inductor, there is energy stored in the magnetic field. Considering a pure inductor L, the instantaneous power which must be supplied to initiate the current in the inductor is :

so the energy input to build to a final current i is given by the integral :

Energy Stored in magnetic field

For the magnetic field the energy density is :

which is used to calculate the energy stored in an inductor

From analysis of the energy stored in an inductor :

the energy density (energy/volume)

is:

so the energy density stored in the magnetic field is

Maxwell equations (Integral form)

Static cases Electric field, (E-field)

electric flux, . electric flux density, D. electric flux intensity, E. permittivity, .

Magnetic field, (H-field). magnetic flux, . magnetic flux density, B. magnetic flux intensity, H. permeability, . Flux linkage,

Electric Variables and Units

Field intensity, E (V/m) Flux density, D (C/m2) Flux, Y (Coulombs) (C) Charge, Q (C) Line charge density, l (C/m) Surface charge density, s (C/m2) Volume charge density, r (C/m3) Capacitance, C (Farads) (F) C = Q / V (F)

Electric field intensity The force experience by any two charged bodies is given by Coulomb’s

law. Coulomb’s force is inversely proportional to the square of distance.

1 22

0

( )4 r

Q QF a Newtons

rpe=r r

( / ) ( / )F

E N C or V mq

=

rr

•Electric field intensity, E, is defined in terms of the force experienced by a test charge located within the field.•We could solve all of the electrostatics with this law as:

,

,

,

2 ,0 ,

1 ( )( )

4 m mall m mspaces

mE m a dV

r

rpe

= òòòr r

Gauss’s Law(Basic electrostatic form)

. enc

s

D dS Qj = =òrr

Ñ

Electric flux is equal to the charge enclosed by a closed surface.• The closed surface is known as a Gaussian surface• Integrate the flux density over the Gaussian surface to calculate the flux.• The flux does not depend on the surface! Use the right one!

The enclosed charge can be eithera point charge

ora charge density such as, , orl s r

Flux density

20 ( / )rB H Wb m Teslamm=

r r

• Magnetic flux density is equal to the product of the permeability and themagnetic field intensity, H.• Magnetic flux density can simplify to flux divided by area.

( )B TA

f=r Equation opposite assumes flux density is

uniform across the area and aligned withthe unit normal vector of the surface!

B = Magnetic flux density (T)0 = Permeability of free space = 4x(10)-7 (H/m)r = Relative permeability of magnetic materialH = Magnetic field intensity (A/m) = Magnetic flux (Wb)

Maxwell’s Static Fields

. 0C

E dl =òr r

Ñ.

s v

D dS dvr=ò òrr

Ñ Ñ

. .C s

H dl J dS=ò òrr r r

Ñ

Electrostatic

Maxwell’s 1st Equation = Faraday’s Law

Maxwell’s 3rd Equation = Gauss’s LawNo Electric fields without charges

Magnetostatics

. 0s

B dS=òrr

Ñ

Maxwell’s 2nd Equation = Ampere’s LawNo magnetic Field without currents

Maxwell’s 4th Equation = Conservation ofMagnetic Flux

Integral Form of Maxwell Equations

Gauss' law for electricity

Gauss' law for magnetism

Faraday's law of induction

Ampere's law

The electric flux out of any closed surface is proportional to the total charge enclosed within the surface

The net magnetic flux out of any closed surface is zero.

The line integral of the electric field around a closed loop is equal to the negative of the rate of change of the magnetic flux through the area enclosed by the loop

In the case of static electric field, theline integral of the magnetic field around a closed loop is proportional to theelectric current flowing through the loop.