MUST KNOW - Electrostatics
About how fast do charges redistribute on a metallic conductor?
(about 10^-19 seconds)
Charge of an electron
-1.602e-19 Coulombs
Boltzmann's constant
1.38e-23 Joules/Kelvin
1 electron-Volt = ?
1.602e-19 Joules
Speed of light
3 * 10^8 m/s
Avogadro's number
6.022e23 molecules per mole
Permittivity of free space
8.854e-12 Farads/meter
Rest mass of an electron
9.1e-31 kg
Give some examples of relative permittivity of some materials.
Air = 1; Fused quartz = 3.9; Silicon = 11.9; distilled water = 81.
What is an equipotential line?
An equipotential line is where the electric potential is constant so any movement of charge around this line requires no work. Equipotential lines are always perpendicular to the gradient of the potential and orthogonal to the electric field.
What is an insulator? Can an insulator be polarized? More broadly, can any material be polarized? Is this always observable?
An insulator is a substance whose electrons and ions cannot move over macroscopic distances under the influence of an electric field. Insulators are said to be polarized when the presence of an electric field displaces the electrons within a molecule away from their average positions. We we consider the polarizability of insulators, we call them dielectrics. Yes, all substances are polarizable, but the effects of polarizability are only readily observed when the material does not conduct electricity-that is, when it is an insulator.
Determine the relationship between surface charge density on a conductor at equilibrium and the electrostatic field at the surface. How does this change if above the conductor is a dielectric and not free space?
Consider an infinitesimal pillbox-shaped Gaussian surface where the portion in the conductor experiences no electric flux and the sides of the pillbox experience no flux because the electric field tangential to the surface is zero and the height of the pillbox is made infinitesimally small. The net flux through the top portion must equal the charge inside the pillbox such that: n_hat dot D = eps0*E_normal = rho_s (surface charge density). The more general relationship replaces eps0 with the particular dielectric constant for whatever material. This result does not contradict the electric field result for an infinite sheet of charge (which wa rho_s/(2eps0) is because the charge on the conductor actually arises from two systems of charges: the local surface charge and the "source" charge (which can be imagined as another far off negatively charged plate). You can also make a Gauss's law argument. Pg 315 Inan for more information.
What are the units of electric displacement?
Coulombs
What does the vector D represent in electrodynamics?
D is the density of electric displacement or flux density. D is defined as Q/(4pi*r^2) r_hat and has units of Coulombs per meter squared. Note that the electric displacement per unit area, D, depends on the orientation of the area; hence, it is a vector quantity. In simple, isotropic materials: D = epsilon*E.
What's a general statement about how electrostatic forces tend to cause dielectrics to be acted upon?
Dielectrics tend to move towards regions of higher electric fields (because the electric field is non-uniform; i.e. its stronger closer to the source than farther away).
What is divergence?
Divergence is a measure of the source density per unit volume of a vector field. Qualitatively, divergence of a vector field is a scalar quantity and is nonzero only at those points where new flux lines emerge or terminate.
Consider two coaxial, concentric and infinitely long cylindrical shells of surface charges with radii a and b, such that b > a. The total amount of charge per unit length on each cylinder is equal in magnitude and opposite in sign, with the surface charge densities on the inner and outer shells being respectively rho_s and -rho_s(a/b). Find the electric field and the electrostatic potential everywhere (pg 300 Inan).
E(r<a) = 0; E(a<r<b)=rho_s*a/(eps0*r) r_hat; E(r>b) = 0. The electric potential is: Phi(r) = -rho_s*a/eps0 * ln(r/b)
Derive the electric field from a spherical cloud of charge with radius, a, and volume charge density, rho_e, for both r<a and r>a.
E(r>a) = kQ/r^2; E(r<a) = kQr/a^3
Derive the electric field and electric potential from an infinite sheet of charge of surface charge density rho_s.
E(z>0) = rho_s/2eps0 z_hat; E(z<0) = -rho_s/2eps0 z_hat; Phi(z) = - Integral( (E_z z_hat) dot (dz z_hat),infinity,0) which equals infinity! It seems as though the work done to bring a positive point charge from infinity to a point requires infinite energy (work) because the electric field is everywhere constant over an infinite distance. We can, however, evaluate the potential between any two points (instead of infinity) and find: Phi(a-b) = rho_s(a-b)/2eps0.
Derive the electric field at a radius r from an infinitely long line charge.
E_r = lambda/(2pi*eps0*r)
What is electric susceptibility? What are its units?
Electric susceptibility is a dimensionless quantity equal to: (epsilon/epsilon0 - 1).
When can polarization of a material be described by simple mathematical expressions for displacement density? When can it not? What is an example where we cannot use these simple expressions?
Expressions like, D = eps0*E + eps0*X_e*E = eps0*(1 + X_e)*E = eps*E, depend on the linear relationship between the polarization per unit volume P and the electric field E. An important example of nonlinear behavior occurs when the electric field is intense enough to pull electrons completely out of their bound locations, causing dielectric breakdown. It also requires the material to be isotropic, that is, X_e, must be independent of direction of E. If the material is not isotropic, and is instead anisotropic (e.g. crystals), then the electric field in one direction can produce a polarization (and thus D) in another direction. Accordingly, this relationship between D and E must be expressed as a matrix, more commonly called a tensor.
How does the electric field fall off for a dipole versus a point charge?
Far from the dipole the electric field varies as r^-3, in contrast to r^-2 dependence for the field of a point charge and r^-1 for the field of a line charge.
Derive Gauss's law for a differential volume.
For the electrostatic field, the total outward flux through a surface S is equal to the total charge enclosed. For a differential volume element V, we can assume the volume charge density to be constant at all points with infinitesimal volume, so the total charge enclosed is rho*V. Accordingly, we have: div(D) = lim V → 0 (closed-loop line integral (D dot ds)/V) = rho, so that the divergence of the electric flux density at any point is equal to the volume charge density at that point.
State Gauss's law succinctly.
Gauss's law states that the total electric flux out of any closed surface S is a constant and is equal to the total charge in the volume V enclosed by S.
What is ionic polarizability?
In some materials, two different atoms may join together as a molecule by forming a chemical bond. We can think of such molecules as consisting of positively and negatively charged ions, with the Coulomb forces between them serving as the binding force. One example is H2O. Depending on whether or not the electrons are transferred or shared, the bond can be ionic or covalent. H2O is partially ionic and may consist of polar molecules which carry a permanent dipole moment.
What is a conservative field?
In terms of electrostatics, if a charge is moved around in a static electric field in the absence of friction (i.e. losses or dissipation of energy), then no energy is dissipated.
Draw the electric field lines between: two positive, one positive and one negative, and one +4Q positively charged and one -1Q negatively charged points.
Inan Inan pg 268.
What is the dot product of two vectors, A and B?
Inan Inan pg 270
Write the mathematical expression for a conservative field in both differential and integral form.
Inan pg 359
Explain dielectric breakdown.
It is when a sufficiently high electric field is applied such that a dielectric material is suddenly transformed from a good insulator into an extremely good conductor, causing substantial current to flow. An important example of nonlinear behavior occurs when the electric field is intense enough to pull electrons completely out of their bound locations, causing dielectric breakdown.
What is the purpose of the polarizability concept?
It simplifies analysis of electrostatic phenomena in the presence of dielectric materials by relating local molecular electric field concepts (microscopic physical behavior) with macroscopic fields.
What is an expression for displacement density or flux density vector? What assumptions are you making?
Most generally, D = eps0*E + P, hold for any material. For a linear dielectric material that follows: P = eps0*X_e*E, D = eps0*E + eps0*X_e*E = eps0*(1 + X_e)*E = eps*E.
What is a nonpolar material? What's an example?
Nonpolar materials consist of molecules that do not possess a permanent dipole moment; the external field both induces the dipoles and orients them.
What is the polarization vector?
P = Np where N is the atoms per unit volume. So P is the dipole moment per unit volume.
What is an expression for the polarization per unit volume, P?
P = eps0*X_e*E
If electrical permittivity is known, how can you find the polarization?
P = eps0*X_e*E = D - eps0*E = (eps - eps0)*E.
Consider two parallel conducting plates separated by free space. Determine the capacitance.
Page 336 Inan
What is the potential due to an electric dipole?
Phi = (1/4pi*eps0)*[p dot r_hat]/r^2 where p is defined as the electric dipole moment and is equal to: (Qd)z_hat and z_hat dot r_hat = cos(theta).
What a material becomes polarized, where is the net charge?
Polarization does not produce a net charge inside the dielectric. Any interior volume of macroscopic dimensions contains equal amounts of positive and negative charge. However, a net amount of surface polarization charge does appear on the surface of the polarized dielectric. This layer of charge adjacent to the boundary remains unneutralized and appears as polarization charge with surface charge density, rho_s_p. The amount of surface charge is a direction indication of the degree of polarization of a material.
What do we call the stored energy in an electric configuration?
Potential, or electric potential, is a quantitative measure of the work or energy required to move charges from one point to another under the influence of an electrostatic field.
What is electric potential? What are its units?
Potential, or electric potential, is a quantitative measure of the work or energy required to move charges from one point to another under the influence of an electrostatic field. To determine the units of electric potential, we know that potential is a measure of the work: Work = - Integral(F,a,b) where F = qE. So for our formal definition of potential: Work/q = - Integral(E,a,b). This shows us the units of Joules per Coulomb. Also known as Volts.
Consider a uniform dielectric slab of permittivity Epsilon1 partially inserted between the plates of a parallel-plate capacitor of width, w, separation, d, and depth, L. Determine the electrostatic force acting on the dielectric slab when the plates are attached to a battery and maintained at constant potential V0. Draw the field lines and direction of force. How much work would it take to place the dielectric slab completely in the capacitor?
Suppose the portion of the dielectric between the plates is of length x. Using, W_e = ½ Volume Integral(D dot E dv), the stored energy of the system is: W_e = ½ eps0 * (V0/d)^2 (w-x)Ld + ½ Epsilon1 * (V0/d)^2 (x)Ld + W_fringing, where the E field in the region between the plates is assumed to be constant [E = V0/d] and the W_fringing term takes account of the fields fields at both sides, which also store some energy. We now consider a virtual displacement d_x of the dielectric such that more of it is now between the plates. From the above expression for W_e, and noting that Epsilon1 > eps0, this displacement would increase the stored energy. We consider, arbitrarily, the virtual displacement to occur under conditions of constant potential, so that the work done by the field (or the energy provided by the battery, which keeps V0 constant) is given by F * d_x, and equals the increase in stored energy. The force on the dielectric can be found from: F * d_x = d_W_e = -½ eps0 * (V0/d)^2 * d_x * L * d + ½ Epsilon1 * (V0/d)^2 * d_x * L * d + dW_fringing/dx * dx. So, F = ½ E^2(Epsilon1 - eps0) * L * d. Where we note the fringing term does not change with the virtual displacement, as long as the ends of the slab are not too close to the plate edges. The direction of the force is to draw the dielectric slab further into the capacitor. In order to calculate the work required (i.e. the energy that needs to be supplied by the batteries) to place the dielectric slab completely inside the capacitor. We have: W = Integral(F * dx,0,w) = ½ E^2 (Epsilon1 - eps0) * Lwd = ½ Volume Integral(Epsilon1*E dot E dv) - ½ Volume Integral(eps0*E dot E dv) = ½ Volume Integral(eps0*E dot E + P dot E dv) - ½ Volume Integral(eps0*E dot E dv) = ½ Volume Integral(P dot E dv) → which is the additional energy needed to polarize the inserted dielectric.
What is electrical permittivity? What are the units?
Symbol: epsilon, also known as the dielectric constant; equal to: epsilon = epsilon0*(1 + X_e). Typically, it is written: epsilon = epsilonR*epsilon0 where epsilonR is the relative permittivity or the relative dielectric constant.
What is the dielectric constant? What are the units?
Symbol: epsilon, also known as the electrical permittivity; equal to: epsilon = epsilon0*(1 + X_e). Typically, it is written: epsilon = epsilonR*epsilon0 where epsilonR is the relative permittivity or the relative dielectric constant.
What is Boltzmann's constant? What does it represent? --
The Boltzmann constant (kB or k) is a physical constant relating the average kinetic energy of particles in a gas with the temperature of the gas and occurs in Planck's law of black-body radiation and in Boltzmann's entropy formula. 1.38064852 × 10-23 m2 kg s-2 K-1
What are the units of electric field?
The SI unit is Newton per Coulomb, or more commonly, Volt per meter.
What happens when you place a conductor in an electric field?
The boundary conditions E_tangential = 0 and D_normal = rho_s (surface charge density) imply that if a conductor is placed in an externally applied electric field, then (1) the field distribution will be distorted so that the electric field lines are normal to the conductor surface, and (2) a surface charge will be induced on the conductor to support the electric field.
What is a good measure of the conducting ability of a material? Give an example.
The charge rearrangement time which is 10^19 seconds for metallic conductors and, for a good dielectric like fused quartz, about 50 days. This means that the free charge deposited inside quartz stays in place for all practical purposes and can be considered as bound charge.
How would you state mathematically the general property of a conservative field?
The closed path line integral (path around any closed loop) of the electric field is always equal to zero.
What assumptions are inherent in the equation, D = epsilon*E?
The dielectric constant, epsilon, is for a simple, isotropic material and epsilon is constant regardless of the strength of E.
State the divergence theorem.
The divergence integrated over the volume V would give the total outward flux from that volume which equals the outward flux through the surface S enclosing the volume.
What is an electric field?
The electric field is a field of force, or, a force per unit charge from F = qE. The notion of a field is a bit abstract but can be defined as a set of values assumed by a quantity at various points in a region of space at various instants of time.
Does the electric flux density depend on permittivity?
The electric flux density vector D is a measure of electric displacement and is a function only of the electric charge producing it, independent of the type of medium surrounding the charge. In free space, D = eps0 * E, so for a point located at the origin we have, D = Q/(4pi*r^2) r_hat. This shows D does not depend on permittivity.
How is the electrostatic potential defined?
The electrostatic potential at any point P is defined as the work required to move a unit positive test charge from infinity to the point P in the presence of an electrostatic field.
Where is the energy stored in an electrostatic configuration?
The energy is "stored" in the configuration of charges. This energy can be recovered by allowing the charges to return to their original relative positions.
In terms of an electrostatic configuration of charges, what is potential energy?
The energy required to hold the charges in the configuration. This electric energy associated with an assembly of charges is only a function of the final configuration and can be determined by calculating the work required to gather the charges together.
What is a gradient?
The gradient of a scalar function at any point is the maximum spatial rate of change of that function at that point. The gradient is thus a vector quantity because the maximum rate of change must occur in a given direction.
What is a polar material? Give an example.
The molecules of polar materials (e.g. NaCl) have a permanent dipole moment, but the individual molecular dipoles are usually randomly oriented due to thermal agitation. When an applied field is present, the individual dipoles tend to align themselves in the direction of the field.
Describe the difference between polarizability and redistribution of charges on a conductor.
The redistribution of charges on a metallic conductor involves the transfer of a very small percentage of the amount of free charge in a conductor over macroscopic distances. Polarization of dielectrics involves the displacement of one or more electrons per atom over subatomic or microscopic distances.
What is capacitance? What are its units?
The relation between induced charges on a set of conductors and the resulting potentials in their vicinity depends only on the geometric arrangement of the conductors. This constant charge-to-potential ratio of an isolated conductor is its capacitance. The units of capacitance are Coulombs per Volt, or Farads.
What is the polarization per unit volume?
The total polarization of a material may arise due to electronic, ionic, and orientational polarizability, leading to the dipole moment per unit volume. The dipole moment per unit volume needs to account for both the externally applied electric field and the local molecular field. The solid dielectrics, the effects of the fields of adjacent molecules cannot be neglected. The molecular field, Emol, and thus the local field, Eloc, can be calculated by removing the molecule in question, maintaining all other molecules in their time-averaged polarized positions, and calculating the space-average electrostatic field in the cavity previously occupied by the molecule. This leads to: Eloc = E + P/3eps0. We can eventually get to: P = eps0*X_e*E where X_e is the dimensionless electric susceptibility.
Is there an electric field inside conductors? Why? What does this mean?
There is zero electric field inside metallic conductors since otherwise charges would continue to flow. For this reason, the electrostatic potential must be the same throughout the metallic conductor. This means that the entire space occupied by a conductor must be an equipotential volume, and, all the charge, if any, on a conductor must reside entirely on its surface since, if any charge did exist within the body, then by Gauss's law a nonzero field would have to exist in the vicinity of a small Gaussian surface surrounding that point. At equilibrium, the surface charge is distributed in such a way that the total electric field inside the conductor and tangential to its surface is zero.
How would you calculate electrostatic forces on a parallel plate capacitor? Describe in words.
To calculate electrostatic forces, we consider how the energy of the system changes for a small virtual change in geometry. This method, referred to as the principle of virtual work. When calculating electrostatic forces using the principle of virtual displacement, the conductors can be kept isolated (i.e. constant charge) or be maintained at constant potential (i.e. by batteries) during their virtual displacement. The energy change accompanying a displacement, dx, is different according to whether or not energy is available from the batteries. The electrostatic force, however, does not depend on this choice.
Derive Gauss's law
To determine the net outward flux from a surface enclosing a point charge, consider the general surface S (any random surface). Emanating from Q (point charge inside S) are cones of flux, which might cross the surface more than one time. Also we can imagine another spherical surface S' around the point charge Q. The net total flux through the surfaces S and S' are identical because [the net flux out of any closed surface that does not enclose any charge must be zero]. Note that because S is arbitrary, this holds for any shape and size. We calculate the electric flux by integrating the electric flux density, D, over the surface of the enclosing sphere which is equal to Q (the enclosed charge). We can thus conclude that the surface integral of any closed surface of D dot ds is equal to the charge enclosed. It is important to note that D dot ds calculates the electric flux normal to the surface ds.
What is the mathematical expression for potential? Explain each part.
W/q = potential (J/C or Volts) = - Integral(E dot dl,a,b). The minus sign in front of the integral is necessary because the work is done against the field. The dot product accounts for the fact that it takes no work to move the test charge perpendicular to the field.
What's the energy stored in a capacitor?
W_e = ½ CV^2 = ½ QV, where we have used C = Q/V.
Why can we use the principle of superposition for electric field measurements?
We are able to assume linearity simply because, at the macroscopic level, its use produces accurate results in many different experiments and applications involving groups of charges.
Calculate the electrostatic forces on a parallel-plate capacitor in free space with a total charge of +Q in one plate and -Q in the other.
We can visualize the upper plate as being displaced upward by an amount d_y. As a consequence, work must be performed by an amount, -F*d_y, where F is the electrostatic force that must be overcome by the external force. From conservation of energy, this mechanical work must reappear as energy somewhere else, and in this case the stored energy of the field is the only other energy term involved, so it must therefore have increased by an amount determined from: d_W_e = W_e|after - W_e|before = ½ Q^2/C_after - ½ Q^2/C_before where Q = eps0 * E * A, as required by Gauss's law and C = eps0 * A/y for the capacitance of parallel plates. It is important to note fringing fields are neglected. We then can calculate that d_W_e = F*d_y = ½ eps0 * E^2 * A *d_y. Such that, F = -eps0 * E^2 * A / 2.
Formally show that the electric field tangential to a conductor surface should be zero.
We use the conservative property of the electrostatic field to draw a rectangular box around the interface of a conductor and free space. We note that the line integral of the electrostatic field around any closed path is identically zero and we know that the line integral of the portion of the rectangular box that resides inside the conductor must equal zero since the electric field is zero inside a conductor. Thus, the tangential component of the electric field at the surface of a conductor is identically zero. The line integral of the rectangular box is made equal to zero by letting the height of the box go to zero.
What is Coulomb's law? Can you derive it?
You cannot derive Coulomb's law. It is based upon physical observation and is not logically or mathematically derivable from any other concept. Coulomb's law states that the electric force between two point charges Q1 and Q2 is proportional to the product of the two charges Q1Q2 and inversely proportional to the square of the distance between them. In vector form: F12 = -F21 = Q1Q2/(4pi*epsilon0*R^2) * R_hat where R = r2 - r1 being the position vectors for points P1 and P2. R_hat is the unit vector in the direction of R and R is the magnitude of R. Note that R is the vector pointing from P1 to P2. The (1/4pi*eps0) is a proportionality constant with the "4pi" factor included so that a 4pi does not appear in Maxwell's equations.
Name the three so-called Gaussian surfaces. What is special about these geometries? What are they good for?
cylindrical pillbox, cylinder, sphere. Gaussian surfaces take advantage of inherent symmetries such that D is everywhere either tangential or normal to the closed surface, so that D dot ds is either zero or simply D_normal*ds with D_normal = constant on the surface. Gauss's law is then useful only for symmetric charge distributions, which lead to symmetric distributions of E and D.
State Gauss's law in terms of both the electric flux density and the electric field.
div(D) = rho → div(E) = rho/eps0.
What is electronic polarizability?
p = alpha*E where alpha = 4pi*eps0*a^3 where a is the effective atomic radius.
Draw Coulomb's law
pg 247 of Inan Inan (must include: r1, r2, R, Q1, Q2, F1, F2, and the origin)
Draw an electric field measured at point P from a test charge Q.
pg 258 Inan Inan
Can you derive the gradient form of the relationship between E and the potential? (hint: make a work argument)
pg 276 Inan
Write E = grad(phi) in cylindrical coordinates.
pg 278 Inan
Write E = grad(phi) in spherical coordinates.
pg 278 Inan
Derive the proper differential expression for the divergence of D
pg 302 Inan
Derive the differential expression for divergence in cylindrical coordinates.
pg 307 Inan
Derive the differential expression for divergence in spherical coordinates.
pg 309 Inan
Find the electric potential and electric field between two concentric cylinders of radius, a and b, where the inner cylinder is at V0 potential and the outer cylinder is at zero. The region between the plates is free space and contains no free charge. Then, find the surface charge densities induced on the two plates.
pg 329 Inan
Find the electric potential and electric field between two parallel plates separated a distance, d, and where the bottom plate is at zero potential and the top plate is at V0. The region between the plates is free space and contains no free charge. Then, find the surface charge densities induced on the two plates.
pg 329 Inan
Find the electric potential and electric field from a spherical conductor maintained at a potential of Phi = V0 by means of a battery. What assumptions do you need to make?
pg 333 Inan
Consider the coaxial capacitor, consisting of two concentric cylinders separated by free space. Find the capacitance per unit length.
pg 337 Inan
Consider a spherical capacitor consisting of two concentric conducting spheres. Find the capacitance.
pg 338 Inan
Derive the electric dipole moment per atom. What are the units of electric dipole moment per atom?
pg 346 Inan; p units are Coulomb-meter-per atom
Derive the charge distributions for a volume distribution of polarization.
pg 350 Inan
Derive the displacement density or flux density vector form of Gauss's law.
pg 351
Write the mathematical expression for Coulomb's law in both differential and integral form.
pg 359 Inan
Explain how D and E changes across a two dielectric interface.
pg 360 Inan The normal component of D is continuous across a charge free dielectric boundary. Then, the normal component of E is then not continuous.
Derive Poisson's equation from Gauss's law.
start with Gauss's law, substitute in E=-grad(V) and perform the scalar product to get the Laplacian of electric potential equals negative the charge density divided by the permittivity.
Derive Laplace's equation from Gauss's law.
substitute E=-grad(V) into Gauss's law.
What is permittivity?
the ability of a substance to store electrical energy in an electric field.
Define a Coulomb
the quantity of charge that flows through a given cross section of wire in one second when there is a steady current of one ampere (A) flowing in the wire.
