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engineering
modern electrodynamics

Questions and Answers of Modern Electrodynamics

Students are often told that E = Fq/q defines the electric field at a point if Fq is the measured force on a tiny charge q placed at that point. More careful instructors let q → 0 to avoid the
A uniform electric field E0 exists throughout a homogeneous dielectric with permittivity ε. What is the electric field inside a vacuum cavity cut out of the interior of the dielectric in the shape
Application 6.3 modeled a meson (a quark-antiquark pair) as a finite dipole placed at the center of a spherical cavity with radius R and unit dielectric constant scooped out of an infinite medium
An origin-centered sphere with permittivity ε and radius a is placed in a uniform external electric field E0. What radius b < a should an origin-centered, perfectly conducting sphere similarly
Two concentric, spherical, conducting shells have radii R2 > R1 and charges q2 and q1. The volume between the shells is filled with a linear dielectric with permittivity ε = κε0. Determine the
The parallel-plate capacitor shown below is made of two identical conducting plates of area A carrying charges ± q. The capacitor is filled with a compressible dielectric solid with permittivity ε
A spherical conductor of radius R1 is surrounded by a polarizable medium which extends from R1 to R2 with dielectric constant κ.(a) The conductor has charge Q. Find E everywhere and confirm that the
A metal ball with charge Q sits at the center of a thin, spherical, conducting shell. The shell has charge Q' and the space between the shell and the ball is filled with matter with dielectric
A dielectric body with permittivity εin is embedded in an infinite volume of dielectric matter with permittivity εout. The entire system is polarized by an external electric field Eext. If ϕ is
Write the Helmholtz theorem expression for D(r) and eliminate D itself from the integrals you write down. How does this formula simplify (if at all) for simple dielectric matter?
The text proved that the force on an isolated dielectric is Where E(r) is the total field at an interior point r and Eavg(rS) is the average of the total field just inside and just
Point charges q1, q2, . . ., qN are embedded in a body with permittivity κin. The latter is itself embedded in a body with permittivity κout. Find the total polarization charge Qpol induced on the
The figure shows two fixed-potential capacitors filled with equal amounts of two different types of simple dielectric matter. Use the stress tensor method to compare the force per unit area which
Use the method of Lagrange multipliers to show that, among all functions D(r) which satisfy ∇ · D = ρf, the minimum (not merely the extremum) of:Occurs when D(r) = εE(r). Assume that ˆn · D
a. Let (ˆe1, ˆe2, ˆe2) be unit vectors of a right-handed, orthogonal coordinate system. Showthat the Levi-Civit`a symbol satisfiesb. Prove thatc. Prove that: d. In quantum mechanics, the
A test function as part of the integrand is required to prove any delta function identity. With this in mind:(a) Prove that δ(ax) = 1/|a|δ(x), a ≠ 0.(b) Use the identity in part (a) to prove
Evaluate the following expressions which exploit the Einstein summation convention.(a). δii.(b). δij εijk.(c). εijk εℓjk.(d). εijk εijk.
Use the Levi-Civit`a symbol to prove that(a) (A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C).(b) ∇ · (f × g) = g · (∇ ×f) − f · (∇ ×g).(c) (A × B) × (C × D) = (A · C ×
Use the Levi-Civita symbol to prove that.(a) ∇ · (f g) = f ∇ · g + g · ∇f.(b) ∇ ×(f g) = f ∇ ×g − g×∇f.(c) ∇ ×(g × r) = 2g + r ∂g/∂r − r(∇ · g).
Without using vector identities:(a) Use Stokes’ Theorem ∫ dS · (∇ ×A) = ∮ ds · A with A = c × F where c is an arbitrary constant vector to establish the equality on the left
(a). Show that δ(r)/r = −δ’ (r) when it appears as part of the integrand of a three-dimensional integral in spherical coordinates. Convince yourself that the test function ∫ (r) does not
Show that D(x) = limm→∞ sin mx/πx is a representation of δ(x) by showing that ∫∞−∞ dxf (x)D(x) = ∫ (0).
Without using vector identities, prove that:(a) ∇f (r − r’) = −∇’ ∫ (r − r).(b) ∇ · [A(r) × r] = 0.(c) dQ = (ds · ∇)Q where dQ is a differential change in Q and ds is an
Assume that ϕ(r) and |G(r)| both go to zero faster than 1/r as r →∞.(a) Let F = ∇ϕ and ∇ · G = 0. Show that ∫ d3r F · G = 0.(b) Let F = ∇ϕ and∇ ×G = 0. Show that ∫ d3r F × G =
Compute the unit normal vector ˆn to the ellipsoidal surfaces defined by constant values ofCheck that you get the expected answer when a = b = c. ①(x, y, z) = V + 6² 2
Express the following in terms of ˆr, ˆ θ, and ˆφ: ar де ar аф əô де a аф аф 20 аф
Let E be a positive real number. Evaluate(a)(b) (c) 81(E)= f dk, 8(E – k²).
Mimic the proof of Helmholtz’ theorem in the text and prove that y(r) = -V. 4π d³r' V'y(r') [r - r' + V 1+1 7 Sa Ал S ds'. (r') r-r'
Derivatives of exp(ik · r) Let A(r) = c exp(ik · r) where c is constant. Show that, in every case, the replacement ∇ →ik produces the correct answer for ∇ · A, ∇ ×A, ∇ ×(∇ ×A),
Let b be a vector and ˆn a unit vector.(a) Use the Levi-Civit`a symbol to prove that b = (b · ˆn) ˆn + ˆn × (b × ˆn).(b) Interpret the decomposition in part (a) geometrically.(c) Let ω = a
A vector function Z(r) satisfies ∇ · Z = 0 and ∇ ×Z = 0 everywhere in a simply-connected volume V bounded by a surface S. Modify the proof of the Helmholtz theorem in the text and show that
Sij and Tij(a) What property must Sij have if εijk Sij = 0?(b) Let b and y be vectors. The components of the latter are defined by yi = bkTki where Tij = −Tji is an anti-symmetric
Let S be the surface that bounds a volume V . Show that (a) ∫s dS = 0;(b) 1/3 ∫s dS · r = V.
If a and b are constant vectors, ϕ(r) = (a × r) · (b × r) is the electrostatic potential in some region of space. Find the electric field E = −∇ϕ and then the charge density ρ = ε0∇
Let A and B be vectors. Show that 1 1 = = €ijk (A × B)k + − (A; B; + A;B;). 2 A; B; =
Let F1 and F2 be the instantaneous forces that act on a particle with charge q when it moves through a magnetic field B(r) with velocities υ1 and υ2, respectively. Without choosing a coordinate
The electric and magnetic fields for time-independent distributions of charge and current which go to zero at infinity are(a) Calculate ∇ · E and∇ ×E.(b) Calculate ∇ · B and∇ ×B. The curl
Let r1 (r2) point to a line element ds1 (ds2) of a closed loop C1 (C2) which carries a current I1 (I2). Experiment shows that the force exerted on I1 by I2 is(a) Show that(b) Use (a) to show that
The magnetostatic equation ∇ × B = μ0j is not consistent with conservation of charge for a general time-dependent charge density. Show that consistency can be achieved using ∇ × B = μ0j + jD
A particle with charge q is confined to the x-y plane and sits at rest somewhere away from the origin until t = 0. At that moment, a magnetic field B(x, y) = Φδ(x)δ(y)Ẑ turns on with a value of
Consider a collection of point particles fixed in space with charge density is :Suppose that : E(r, t = 0) = B(r, t = 0) = 0 and(a) Construct a simple current density which satisfies the continuity
Let θ be a parameter and define “new” electric and magnetic field vectors as linear combinations of the usual electric and magnetic field vectors:E' = E' cos θ + cBsin θcB' = −E sin θ + cB
An infinitely long cylindrical solenoid carries a spatially uniform but time-dependent surface current density K(t) = K0(t/τ). K0 and τ are constants. Find the electric and magnetic fields
If α is a real constant, the continuity equation is satisfied by the charge and current distributions. The given j represents current flowing in toward the origin of coordinates. But the given ρ is
A point particle with charge q and mass m is fixed at the origin. An identical particle is released from rest at x = d. Find the asymptotic (x →∞) speed of the released particle.
A surface current density K(rS, t) flows in the z = 0 plane which separates region 1 (z > 0) from region 2 (z < 0). Each region contains arbitrary, time-dependent distributions of charge and
In 1942, Boris Podolsky proposed a generalization of electrostatics that eliminates the divergence of the Coulomb field for a point charge. His theory retains ∇ ×E = 0 but replaces Gauss’ law
If the photon had a mass m, Gauss’ law with E = −∇ϕ changes from ∇2ϕ = −ρ/ ε0 to an equation which includes a lenght L = − ђ/mc:Experimental searches for m use a geometry first
Suppose that the electric potential of a point charge at the origin wereEvaluate a superposition integral to find ϕ inside and outside a spherical shell of radius R with uniform. Check the η → 0
(a) A point charge q > 0 with total energy E travels through a region of constant potential V1 and enters a region of potential V2 < V1. Show that the trajectory bends so that the angles θ1
The Cartesian components of the electric field in a charge-free region of space are Ek = Ck + Djkrj, where Ck and Djk are constants.(a) Prove that Djk is symmetric (Djk = Dkj) and traceless
Draw the electric field line pattern for a line of five equally spaced charges with equal magnitude but alternating algebraic signs, as sketched below.Be sure to choose the scale of your drawing and
Use Gauss’ law to find the electric field when the charge density is:(a) Expresses the answer in Cartesian coordinates.(b) Express the answer in cylindrical coordinates.(c) Express the answer in
Show that the torque exerted on a charge distribution ρ(r) by a distinct charge distribution ρ' (r') is N = 1 471060 [a³r [d²³₁ farfar rxr \r— r'/³0 (r)p'(r'′).
The z-axis coincides with the symmetry axis of a flat disk of radius a in the x-y plane. The disk carries a uniform charge per unit area σ < 0. The rim of the disk carries an additional uniform
(a) Find E(r) if ρ(x, y, z) = σ0δ(x) + ρ0θ(x) − ρ0θ(x − b).(b) Show by explicit calculation that ρ(x, y, z) does not exert a net force on itself.
A charge distribution ρ(r) with total charge Q occupies a finite volume V somewhere in the half-space z < 0. If the integration surface is z = 0, prove that z=0 dS 2. E = ala 으. 2€0
(a) Use Green’s second identity, ∫V d3r (f∇2g − g∇2f ) = ∫S dS · (f∇g − g∇f), to prove that the potential ϕ(0) at the center of a charge-free spherical volume V is equal to
A two-dimensional disk of radius R carries a uniform charge per unit areaσ > 0.(a) Calculate the potential at any point on the symmetry axis of the disk.(b) Calculate the potential at any point
The figure below shows a circular hole of radius b (white) bored through a spherical shell (gray) with radius R and uniform charge per unit area σ.(a) Show that E(P) = (σ/2ε0)[1 −
The figure below shows a cube filled uniformly with charge. Determine the ratio ϕ0/ϕ1 of the potential at the center of the cube to the potential at the corner of the cube. -% S
Suppose the electrostatic potential of a point charge were ϕ(r) = (1/4πε0)r −(1+ε) rather than the usual Coulomb formula.(a) Find the potential ϕ(r) at a point at a distance r from the center
Let the space between two concentric spheres with radii a and R ≥ a be filled uniformly with charge.(a) Calculate the total energy UE in terms of the total charge Q and the variable x = a/R.
(a) Evaluate the relevant part of the integral to find the interaction energy VE between two identical insulating spheres, each with radius R and charge Q distributed uniformly over their surfaces.
(a) Show that the electric field produced by a uniform charge density ρ confined to the volume V enclosed by a surface S can be written(b) Show that the electric field due to an arbitrary but
Suppose that the electrostatic potential produced by a point charge was not Coulombic, but instead varied with distance in a manner determined by a specified scalar function f (r):(a) Calculate the
Let d and s be two unequal lengths. Assume that charge is distributed on the z = 0 plane with a surface density(a) Integrate σ to find the total charge Q on the plane.(b) Show that the potential
(a) The potential takes the constant value ϕ0 on the closed surface S which bounds the volume V. (b) The total charge inside V is Q. There is no charge anywhere else. Show that the
A common biological environment consists of large macro-ions with charge Q < 0 floating in a solution of point-like micro-ions with charge q > 0. Experiments show that N micro-ions adsorb onto

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