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Lecture_4_1.pptx4. Electrostatics
Applied EM by Ulaby, Michielssen and Ravaioli
Maxwell’s Equations
Maxwell’s Equations:
On the left are Maxwell’s Equations (written in differential form). Don’t be scared, just be one with the equations for a little bit. These four partial differential equations form the fundamental tenants of classical electromagnetic theory. Everything we have to say regarding the electric and magnetic fields stem from these four relatively simple equations. They tell us what creates the fields and how the fields evolve in time. They tell us that visible light itself is an electromagnetic wave. These equations are our torch in the cave of the invisible.
So, what are all these symbols in the equation?
J = current density
Rho_v = charge density
H = magnetic field strength
B = magnetic flux density
E = electric field strength
D = dielectric flux density
While it looks like there are four variables to describe the field (E,D, H, and B), E is related to D and H is related to B via constitutive relations. In linear media, like free-space we have D = epsilon*E and B = mu*H
Notice that these equations involve the time-rate of change. As such they govern the dynamics of the fields (i.e. how they change in time). For example, the second equation (which is Faraday’s Law) shows that the curl of the electric field is equal to the negative of the time rate of change of the magnetic B field.
Now, by the way, there is a theorem called the “helmholtz decomposition” that basically says Any sufficiently smooth vector field is uniquely defined if its curl and divergence are known, and if the field vanishes at infinity. Therefore, you would expect a general a general law involving vector quantities E and B to look like (del dot E = ... Del cross E = Del dot B = and del dot H =).
For the next few lectures, we will assume we are in a system where the fields are static (i.e. not changing in time). In this case, we can set the terms with the time derivatives to zero. This results in the equations here on right. The first set here, which govern electrostatics, involve only the electric field E and electric flux density D (not magnetic field). The second set, which govern magneto statics, involve the magnetic field B and magnetic intensity H (no electric field).
These static equations show that the electric fields don’t influence the magnetic fields (and vice versa), Another way of saying this is that the electric and magnetic fields are uncoupled. Therefore we can study electric phenomena separately from magnetic phenomena. This lecture covers electrostatics. In a later lecture we will consider magnetostatics. And then we’ll eventually consider the time-varying case, in which the electric and magnetic fields are coupled and governed by the dynamic and general form of Maxwell’s equations.
Historically, people thought of electrostatics and magnetostatics as two completely separate physical phenomena (I mean, it seems reasonable to assume that permanent magnets and lightning bolts are two totally different things). However, eventually people realized that electric and magnetic fields are in fact coupled, which eventually gave rise to the concept of the “electromagnetic” field.
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Charge Distributions
Volume, surface, and line charge densities:
In electrostatics, electric fields are produced by charges. Charged particles are fundamentally discrete (think about ions, electronics, protons). However, the size of the charged particles themselves are so small compared to distances we typically care about, so that we can model the massive collection of discrete charges as a continuous charge distribution.
These charge distributions can be distributed over a volume, a thin surface, or a thin line in space.
For volumetric charge distributions, we define a volume charge density rho_v. imagine you take a small volume delta_V and measure the amount of charge contained in the volume delta_q. rho_v is equal to delta_q/detla_V inthe limit that the volume goes to zero. The units are in [C/m^3].
For surface charge distributions, we define a surface charge density rho_s. imagine you take a small surface delta_S and measure the amount of charge contained on the surface delta_q. rho_S is equal to delta_q/detla_S inthe limit that the volume goes to zero. The units are in [C/m^2].
In a similar manner you can definite a line charge density rho_l which has units [C/m]
3
Example of charge distribution
Calculate the total charge Q contained in a thin cylindrical tube orientated along the z-axis. The line charge density is
where z is the distance in meters from the bottom end of the tube. The tube length is 10 cm.
[[[READ AND DO EXAMPLE]]]
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Example of charge distribution
A solid half-sphere with radius R = 2 m has a constant volume charge density of
Calculate the total charge Q contained in the half-sphere.
[[[READ AND DO EXAMPLE... Show the integral way and the “easy” way by multiplying volume of half sphere with rho_v]]]
5
Coulomb’s Law
Electric field at point P due to single charge
Electric force on a test charge placed at P
Electric flux density D
As we’ve seen earlier, Coulomb’s Law gives the electric field for a point charge. Suppose a point charge q is at the origin. The electric field at point P is given by this equation. Here R is the straight line distance between the charge q and the point P. R^ is a unit vector which points from the charge q to point P along this line, as indicated in the figure.
In this equation epsilon is the electric permittivity of the surrounding medium.
The electric force on a test charge q’ placed at P would be subjected to a force equal to q’*E.
As we’ve seen earlier, the electric flux density D is equal to E by this equaiton D = eps*E. Note that D is independent of the electric permittivity.
[[[READ LITTLE BOX ON BOTTOM RIGHT]]]
It will be shown shortly that Coulomb’s Law can be derived Maxwell’s Eauaitons (in particular, it willl be derived from Gauss’s Law, which is one of Maxwell’s Equations).
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Coulomb’s Law: Charge not at origin
What if the charge is not at the origin?
[[[GENERALIZE COULOMB’S LAW BY DRAWING SITUATION ON BOARD]]]
Position vector to q is R1
Position vector is P is R
Vector from q to P is R-R1
Distance from q to P is |R-R1|
Unit vector from q to P is (R-R1)/(|R-R1|)
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Electric Field Due to 2 Charges
[[[READ SLIDE]]]
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Electric Field due to Multiple Charges
[[[READ AND DO EXAMPLE]]]
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P(0,0,h)
[[[READ AND DO EXAMPLE]]]
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