Lecture 4 - Current Density, Conductors and Capacitance

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Engineering Electromagnetics

LECTURE 4: Current Density, Conductors and Capacitance

Instructor: Sir Tim

Summer Term

Current Density, Conductors and Capacitance


CURRENT AND CURRENT DENSITY


Current

The term current refers to the rate at whichpositive charges flow.

=


Current Density

However for field theory, the amount ofcurrent per unit area, referred to as thecurrent density is much more useful.

=


Drift Velocity

Refers to the flow velocity the chargeexperience due to applied electric field.

=


J and E Relationship

Drift velocity is a quantity influenced by thestrength of the electric field intensity, i.e.

Hence, the current density can be obtainedby:

=


Resistance

Consider a conductor with uniform surface,the current is

= =

And electric potential

=

=

=


Resistance

Getting the ratio of these two quantities, wehave:

=

=

=


Problem 1

Given the vector current density

J = 102za - 4cos2a mA/m

2

a) Find the current density at P(3, 30o, 2)

b) Determine the current flowing outwardthrough a circular band = 3, 0< < 2,2< z < 2.8.


Problem 2

A wire of diameter 1 mm and conductivity 5x 107 S/m has 1029 free electrons/m3 whenan electric field 10 mV/m is applied.Determine:

a) The charge density of free electrons

b) The current density

c) The current in the wire

d) The drift velocity


Problem 3

An AWG #12 copper conductor has an 80.8mil diameter. A 50-foot length carries acurrent of 20 A. Find the electric fieldintensity E, drift velocity U, the voltagedrop, and the resistance for the 50 footlength. (Cu = 5.8 x 10

7 S/m, Cu = 0.0032m2/V-s)



CURRENT CONTINUITY EQUATION


Current Continuity Equation

The time rate of decrease of charge within agiven volume must be equal to the netoutward current flow through a closedsurface of the volume

= =

=


Problem 1

Current density is given in cylindricalcoordinates as J = 106z1.5az A/m

2 in theregion 0 20 m; for 20m, J = 0.

a) Find the total current crossing thesurface z = 0.1 m in the az direction.

b) If the charge velocity is 2 106 m/s at z= 0.1 m, find v there.

c) If the volume charge density at z = 0.15m is 2000 C/m3, find the chargevelocity there.



CONDUCTORS


Conductors

A conductor is a material that has anabundance of charge carriers (free electrons)


Conductor Boundary Conditions


Problem 1

Given the potential,

= 100(2 2)

and a point P(2,1, 3) that is stipulated tolie on a conductor-to-free-space boundary,find V, E, D, and S at P.


Method of Images


Problem 2

Find the surface charge density at P(2, 5, 0)on the conducting plane z = 0 if there is aline charge of 30 nC/m located at x = 0, z =3



CAPACITANCE


Capacitance

Capacitance measures the capability of energy storage in electrical devices


Parallel Plate Capacitor


Problem 1

Calculate the capacitance of a parallel-platecapacitor having a mica dielectric, r = 6, aplate area of 10 in.2, and a separation of 0.01in.


Problem 2

Find the relative permittivity of thedielectric material present in a parallel-platecapacitor if:

a) S = 0.12 m2, d = 80 m, V0 = 12 V, andthe capacitor contains 1 J of energy

b) the stored energy density is 100 J/m3,V0 = 200 V, and d = 45 m

c) E = 200 kV/m and S = 20 C/m2


Problem 3

Two concentric spherical conducting shells areseparated by vacuum. The inner shell has totalcharge +Q and outer radius ra and the outer shellhas charge Q and inner radius rb. Find thecapacitance of this spherical capacitor.


Problem 4

Show that for a coaxial cable transmission linewith length L, the distributed capacitance(capacitance per unit length) is given by:

/ =2

ln

where : rb and ra are the outer and inner radiusof the coaxial cable respectively



CAPACITORS IN SERIES AND PARALLEL


Capacitors in Series

1 2 3

1 2 3

11 1 1

1 2 3

...

...

...

T

T

T

Q Q Q Q

V V V V

C C C C


Capacitors in Parallel

1 2 3

1 2 3

1 2 3

...

...

...

T

T

T

V V V V

Q Q Q Q

C C C C


Problem 1

For the combination of capacitances shown,assume C1 = 12.0 F, C2 = 5.30 F, and C3= 4.50 F. If V = 12.5 V, What is thecharge on C1?


Problem 2

Derive an expression for the capacitance of aparallel plate capacitor containing twodielectrics.



SEATWORK


Thank you.

Lecture 4: Current Density, Conductors and Capacitance

Engineering Electromagnetics

Timothy M. AmadoFaculty, Electronics Engineering Department

Technological University of the Philippines Manila

[email protected]

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Lecture 4 - Current Density, Conductors and Capacitance