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engineering
modern electrodynamics

Modern Electrodynamics 1st Edition Andrew Zangwill - Solutions

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 of a rectangular pancake with dimensions L × L × h? Assume that h << L and express Ecav
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 with dielectric constant κ → 0. For this problem, we replace the finite dipole by a point dipole
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 placed in a similar field E0 have so that the two situations produce identical electric fields for r
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 elements of the capacitance matrix for this system.
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 and elastic energy(a) Find the equilibrium separation between the plates d(q).(b) Sketch the
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 total polarization charge is zero.(b) The conductor is grounded and the entire system is placed in
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 constant κ. Use the stress tensor method to prove that if the shell were split into two hemispheres, the
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 the exact electrostatic potential and S is the surface of the embedded body, show that the dipole
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 outside the dielectric at the surface point rS. Show that this expression can be rewritten in the form:
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 boundary between the two dielectrics.
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 acts on the two dielectric interfaces. Express your answer in terms of the electric field E0 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 takes specified values on the boundary surface of the volume V of integration. UE=/fd³r |D|²/€
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 Cartesian components of the angular momentum operator ˆL obey the commutation relation. Let a and b be
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 that(c) Confirm that 8[g(x)] = [ m 1 [g'(xm)| 8(x - xm), where g(x) = 0 and g'(xm) 0.
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 × D)B − (B · C × D)A.(d) The 2 × 2 Pauli matrices σx, σy , and σz used in quantum mechanics
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 side of(b) Confirm the equality on the right side of this expression.(c) Show that ∮c r × ds = 2
(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 provide any information. Then try ∫ (r)/r.(b). Show that ∇ · [δ(r − a)ˆr] = (a2/r2)δ’
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 element of arc length.
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 = 0.(c) Begin with the vector with components ∂j (PjG) and prove that [d³r² = -
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), ∇(∇ · A), and ∇2A.
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 · (b × c) where a, b, and c are any three non-coplanar vectors. Now letExpress Ω = A · (B × C)
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 Z(r) can be found everywhere in V if its value is specified at every pointon S.
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 object. Find a vector ω such that y = b × ω. Why does it makes sense that T and ωcould have the
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∇ · E associated with this potential.
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 system, show that B(r) can be determined from the observables υ1 × F1 and υ2 × F2 if υ1 and υ2
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 calculation exploits the continuity equation for this situation. E(r) = 1 477 60 d³r'
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 F₁ = - фис Ijds₁ $12ds2 Mo 4л C1 C2 11 - 12 |r₁ r₂|³ -
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 and a suitable choice for 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 Φ which increases at a constant rate from zero. During the subsequent motion of the particle, show
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 equation.(b) Find B(r, t) and show that this field and E(r, t) satisfy all four Maxwell equations.(c)
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 cos θ.(a) Show that E' and B' satisfy the Maxwell equations without sources (ρ = j = 0) if E and B
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 everywhere in space.
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 translationally invariant, i.e., it does not distinguish any origin of coordinates. Resolve this
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 current.(a) For fields B(r, t) and E(r, t), use the theta function method of the text to derive a
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 by(a) Find the electric field predicted by this equation for a point charge at the origin by writing E
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 employed by Cavendish where a solid conducting sphere and a concentric, conducting, spherical shell
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 limit. φ(r) = q 1 Απερρίτη €0 0 < n < 1.
(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 and θ2 in the diagram below obey a type of “Snell’s law” with a characteristic “index of
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 (∑k Dkk = 0).(b) Find the most general electrostatic potential that generates the electric field in
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 the number of lines drawn so all salient features of the pattern are obvious. + 1 + 1 +
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 spherical coordinates. : Po exp{-K√x p(x) = po exp
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 charge per unit length λ > 0. Use a side (edge) view and sketch the electric field lines
(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 the average of ϕ(r) over the surface S of the sphere. We proved this theorem in the text using
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 on the rim of the disk.(c) Sketch the electric field pattern everywhere in the plane of the disk.(d)
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 − sin(θ0/2)]ˆr, where P is the point at the center of the hole and θ0 is the opening angle of a cone whose apex
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 of a spherical shell of radius R > r with uniform surface charge per unit area σ. Check the
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. Check the a = 0 and a = R limits.(b) Minimize UE with respect to x (keeping the total charge Q
(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. The center-to-center separation between the spheres is d > 2R. Do not assume that d>>R.(b)
(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 localized charge distribution can always be written in the form E(r) = P Απερ J S ds' |r – r|
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 potential produced by an infinite flat sheet at z = 0 with uniform charge per unit area σ.(b) Use
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 ϕ(z) produced by σ(ρ) on the z-axis is identical to the potential produced by a point with charge Q
(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 electrostatic energy contained in the space outside of S is 1 UE (out) == 290.
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 the surface of each macro-ion. Model one macro-ion as a sphere with its charge uniformly distributed

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