1. What does the Green function G of the wave equation in three-dimensional free space represent

physically?

(a) a planar standing wave due to a sheet source

(b) a planar traveling wave due to a point source

(c) a spherical traveling wave due to a point source

(d) the static fields due to a point source

2. Starting with Maxwell's equations in complete form, when is the potential formulation E=−∇ Φ

valid?

(a) always

(b) only when the magnetic field varies very slowly compared to the system (or is constant in time)

(c) only when the magnetic field varies very quickly compared to the system

(d) only when the currents are oscillating quickly

3. What type of force does a moving charged particle feel due to a converging magnetic field (in

addition to the force component that causes spiraling motion)?

(a) it's attracted to the region of convergence

(b) it's repelled from the region of convergence

(c) it depends on the polarity of the B field convergence and the charge of the particle

(d) no additional force

4. A point charge +q is brought close to another point charge -q and they are both brought close to a

infinite grounded conducting slab that fills the space z < 0. A distance r far away from the charges, how

does the electric field depend on r?

(a) 1/r

(b) 1/r2

(c) 1/r3

(d) 1/r4

5. If a unit point charge is on the z axis at x' and the observation point x is at an arbitrary location, when

we make a series expansion of 1/|x-x'| in spherical coordinates, what functions do we use?

(a) Bessel functions

(b) Legendre polynomials

(c) complex exponentials

(d) gamma functions

6. A hollow cylinder with radius a and height L is centered on the z axis with its bottom at z = 0. Its

sides and bottom face are held at an electrostatic potential of zero. Its top face is held at the electrostatic

potential V(ρ) where ρ is the radial coordinate of cylindrical coordinates (note the potential does not

depend on the azimuthal angle). What is the electrostatic potential everywhere inside the cylinder after

all boundary conditions have been applied except the one at the top surface?

(a) Φ(ρ , ϕ , z )=∑ Al P l

l=0

∞

∞

∞

() ( )

ρlπz

sin

aL

(b) Φ(ρ , ϕ , z )= ∑ ∑ J m x mn

m=1 n=1

∞

(

ρzi mϕ−i m ϕ(C m , n e +D m , n e)sin x mn

aa

)

( )

(c) Φ(ρ , ϕ , z )=∑ An N 0 x 0,n

n=1

∞

(

ρz

sin x 0, n

aa

) (

)

)

(d) Φ(ρ , ϕ , z )=∑ An J 0 x 0, n

n=1

(

ρz

sinh x 0,n

7. A cube of uniform paramagnetic material is placed in an originally uniform, external magnetic

field B0 =B0 z . After the system has reached equilibrium, how does the total magnetic ̂�field strength Bin

at the center of the cube compare to the original field strength?

(a) Bin = B0

(b) Bin < B0

(c) Bin > B0

(d) Bin = 0

8. A cylinder is uniform and infinite in the z direction and its surface is held at the electrostatic

potential V (ϕ) where ϕ is the azimuthal angle of cylindrical coordinates. We want the electrostatic

potential everywhere external to the cylinder. In solving the appropriate Laplace equation using the

separation of variables, how many separation constants are there?

(a) 4

(b) 3

(c) 2

(d) 1

9. What does classical electromagnetics tell us about the conservation of electromagnetic energy (when

only linear, non-lossy materials are present)?

(a) energy is conserved, and this fact is contained in Maxwell's equations

(b) energy is conserved, but this is an experimental observation that must be added to Maxwell's

equations

(c) energy is not conserved

(d) energy is not defined in classical electromagnetics

10. A sheet of electrical current exists in the y-z plane and flows uniformly in the z direction such that

J= K 0 δ( x) z . What is the H field everywhere?̂�

11. What is the key mathematical concept that lets us convert ∇⋅D=ρ into a boundary condition?

(a) the divergence theorem

(b) Stokes theorem

(c) Green's theorem

(d) the fundamental theorem of vector calculus

12. Why are B field lines never created or destroyed?

(a) because the effective current always cancels the applied current

(b) because the magnetic monopole moment is always the strongest

(c) because energy is conserved

(d) because there are no real magnetic charges

13. Why do we have to use the retarded charge density in Coulomb's law when the charge density

changes in time?

(a) to make sure radiation travels faster than c

(b) to ensure causality (cause and effect)

(c) to guarantee conservation of charge

(d) to cancel out the infinite field at a point charge

14. Consider an initially static, but non-uniform magnetic field B. How can we make the E field curl?

(a) make the overall field strength of B oscillate in time without changing its spatial pattern

(b) make the spatial pattern of B change continually in time without changing its overall strength

(c) keep the overall strength and pattern of B the same, but shift it continually in space

(d) all of the above

15. A large sphere with a total electrostatic charge +Q fixed permanently and uniformly over its surface

is brought close to an even larger sphere that is conducting and carries a total charge +Q/2. Both

spheres are insulated (not grounded). What is the force between the two spheres?

(a) always repulsive

(b) always attractive

(c) repulsive when they are far away, attractive when they are close enough

(d) attractive when they are far away, repulsive when they are close enough

1. ̂¿Qué ̂hace ̂la ̂función ̂de ̂Green ̂G ̂de ̂la ̂ecuación ̂de ̂onda ̂en ̂el ̂espacio ̂libre ̂en ̂tres ̂dimensiones ̂representanfísicamente?̂(A) ̂un ̂plano ̂de ̂pie ̂de ̂onda ̂debido ̂a ̂una ̂fuente ̂de ̂hoja(B) ̂una ̂onda ̂plana ̂viajar ̂debido ̂a ̂una ̂fuente ̂puntual(C) ̂una ̂onda ̂esférica ̂debido ̂a ̂una ̂fuente ̂puntual(D) ̂los ̂campos ̂estáticos ̂debido ̂a ̂una ̂fuente ̂puntual

2. ̂A ̂partir ̂de ̂las ̂ecuaciones ̂de ̂Maxwell ̂en ̂forma ̂completa, ̂cuando ̂es ̂el ̂potencial ̂formulación ̂E ̂= ̂-∇ ̂Φválido?̂(A) ̂siempre(B) ̂sólo ̂cuando ̂el ̂campo ̂magnético ̂varía ̂muy ̂lentamente ̂en ̂comparación ̂con ̂el ̂sistema ̂(o ̂es ̂constante ̂en ̂el ̂tiempo)(C) ̂sólo ̂cuando ̂el ̂campo ̂magnético ̂varía ̂muy ̂rápidamente ̂en ̂comparación ̂con ̂el ̂sistema ̂de(D) ̂sólo ̂cuando ̂las ̂corrientes ̂están ̂oscilando ̂rápidamente

3. ̂¿Qué ̂tipo ̂de ̂fuerza ̂no ̂una ̂partícula ̂cargada ̂en ̂movimiento ̂se ̂siente ̂debido ̂a ̂un ̂campo ̂magnético ̂convergente ̂(enAdemás ̂de ̂la ̂componente ̂de ̂fuerza ̂que ̂provoca ̂el ̂movimiento ̂en ̂espiral)?̂(A) ̂se ̂siente ̂atraída ̂por ̂la ̂región ̂de ̂convergencia(B) ̂que ̂es ̂repelido ̂por ̂la ̂región ̂de ̂convergencia(C) ̂depende ̂de ̂la ̂polaridad ̂de ̂la ̂convergencia ̂campo ̂B ̂y ̂la ̂carga ̂de ̂la ̂partícula(D) ̂ninguna ̂fuerza ̂adicional

4. ̂Una ̂carga ̂puntual ̂+ ̂q ̂se ̂pone ̂cerca ̂de ̂otra ̂-q ̂carga ̂puntual ̂y ̂ambos ̂se ̂ponen ̂cerca ̂de ̂uninfinita ̂losa ̂realización ̂de ̂tierra ̂que ̂llena ̂el ̂espacio ̂z ̂<0. ̂A ̂distancia ̂r ̂lejos ̂de ̂los ̂cargos, ̂la ̂formano ̂el ̂campo ̂eléctrico ̂depende ̂de ̂r?̂(A) ̂1 ̂/ ̂r(B) ̂1 ̂/ ̂r2(C) ̂1 ̂/ ̂r3(D) ̂1 ̂/ ̂r4

5. ̂Si ̂una ̂carga ̂puntual ̂unidad ̂está ̂en ̂el ̂eje ̂z ̂de ̂x ̂'y ̂el ̂punto ̂de ̂observación ̂x ̂está ̂en ̂una ̂ubicación ̂arbitraria ̂cuando,hacemos ̂un ̂desarrollo ̂en ̂serie ̂de ̂1 ̂/ ̂| ̂x-x ̂'| ̂en ̂coordenadas ̂esféricas, ̂qué ̂funciones ̂usamos?̂(A) ̂las ̂funciones ̂de ̂Besselpolinomios ̂(b) ̂de ̂Legendre(c) ̂exponenciales ̂complejas(d) ̂funciones ̂gamma

6. ̂Un ̂cilindro ̂hueco ̂con ̂un ̂radio ̂y ̂altura ̂L ̂se ̂centra ̂en ̂el ̂eje ̂z ̂con ̂su ̂parte ̂inferior ̂en ̂z ̂= ̂0. ̂Sulados ̂y ̂la ̂cara ̂inferior ̂se ̂mantienen ̂a ̂un ̂potencial ̂electrostático ̂de ̂cero. ̂Su ̂cara ̂superior ̂se ̂lleva ̂a ̂cabo ̂en ̂la ̂electrostáticapotencial ̂V ̂(ρ), ̂donde ̂ρ ̂es ̂la ̂coordenada ̂radial ̂de ̂coordenadas ̂cilíndricas ̂(tenga ̂en ̂cuenta ̂el ̂potencial ̂nodependerá ̂del ̂ángulo ̂azimutal). ̂¿Cuál ̂es ̂el ̂potencial ̂electrostático ̂en ̂todas ̂partes ̂en ̂el ̂interior ̂del ̂cilindro ̂despuéstodas ̂las ̂condiciones ̂de ̂frontera ̂se ̂han ̂aplicado, ̂excepto ̂el ̂de ̂la ̂superficie ̂superior?̂(A) ̂Φ ̂(ρ, ̂φ, ̂z) ̂= ̂Σ ̂Al ̂P ̂ll ̂= ̂0∞∞∞() ̂()ρlπz  pecadoaL(B) ̂Φ ̂(ρ, ̂φ, ̂z) ̂= ̂Σ ̂Σ ̂J ̂m ̂x ̂mnm ̂= ̂1 ̂n ̂= ̂1∞

(ρzi ̂mφ-i ̂m ̂φ ̂(C ̂m, ̂n ̂D ̂e ̂+ ̂m, ̂n ̂e) ̂sen ̂x ̂mnaa)()(C) ̂Φ ̂(ρ, ̂φ, ̂z) ̂= ̂Σ ̂Un ̂N ̂0 ̂x ̂0, ̂nn ̂= ̂1∞(ρz  sen ̂x ̂0, ̂naa) ̂())(D) ̂Φ ̂(ρ, ̂φ, ̂z) ̂= ̂Σ ̂Un ̂J ̂0 ̂x ̂0, ̂nn ̂= ̂1(ρz  senh ̂x ̂0, ̂n

7. ̂Un ̂cubo ̂de ̂material ̂paramagnético ̂uniforme ̂se ̂coloca ̂en ̂un ̂uniforme ̂originalmente, ̂magnético ̂externo              campo ̂B0 ̂= ̂B0 ̂z. ̂Después ̂de ̂que ̂el ̂sistema ̂ha ̂alcanzado ̂el ̂equilibrio, ̂¿cómo ̂la ̂fuerza ̂del ̂campo ̂magnético ̂total ̂Binen ̂el ̂centro ̂del ̂cubo ̂para ̂comparar ̂la ̂intensidad ̂de ̂campo ̂original?̂(A) ̂Bin ̂= ̂B0(B) ̂Bin ̂<B0(C) ̂Bin> ̂B0(D) ̂Bin ̂= ̂0

8. ̂Un ̂cilindro ̂es ̂uniforme ̂e ̂infinita ̂en ̂la ̂dirección ̂z ̂y ̂su ̂superficie ̂se ̂mantiene ̂a ̂la ̂electrostáticapotencial ̂V ̂(φ), ̂donde ̂φ ̂es ̂el ̂ángulo ̂azimutal ̂de ̂coordenadas ̂cilíndricas. ̂Queremos ̂que ̂la ̂electrostáticapotencial ̂en ̂todas ̂partes ̂externa ̂al ̂cilindro. ̂En ̂la ̂solución ̂de ̂la ̂ecuación ̂de ̂Laplace ̂apropiado ̂utilizando ̂elseparación ̂de ̂variables, ̂el ̂número ̂de ̂constantes ̂de ̂separación ̂hay?̂(A) ̂4(B) ̂3(C) ̂2(D) ̂1

9. ̂¿Qué ̂electromagnetismo ̂clásicos ̂nos ̂dicen ̂acerca ̂de ̂la ̂conservación ̂de ̂la ̂energía ̂electromagnética ̂(cuandosólo ̂materiales ̂lineales, ̂sin ̂pérdidas ̂están ̂presentes)?̂(A) ̂la ̂energía ̂se ̂conserva, ̂y ̂este ̂hecho ̂está ̂contenida ̂en ̂las ̂ecuaciones ̂de ̂Maxwell(B) ̂la ̂energía ̂se ̂conserva, ̂pero ̂esta ̂es ̂una ̂observación ̂experimental ̂que ̂debe ̂ser ̂añadido ̂a ̂Maxwell ̂deecuaciones(C) ̂la ̂energía ̂no ̂se ̂conserva(D) ̂la ̂energía ̂no ̂se ̂define ̂en ̂electromagnetismo ̂clásica

10. ̂Existe ̂una ̂hoja ̂de ̂corriente ̂eléctrica ̂en ̂el ̂plano ̂yz ̂y ̂fluye ̂de ̂manera ̂uniforme ̂en ̂la ̂dirección ̂z ̂de ̂tal ̂manera ̂queJ ̂= ̂K ̂0 ̂δ ̂(x) ̂z. ̂¿Cuál ̂es ̂el ̂campo ̂H ̂en ̂todas ̂partes?̂

11. ̂¿Cuál ̂es ̂el ̂concepto ̂matemático ̂clave ̂que ̂nos ̂permite ̂convertir ̂∇⋅D ̂= ̂ρ ̂en ̂una ̂condición ̂de ̂frontera?̂(A) ̂el ̂teorema ̂de ̂la ̂divergencia(B) ̂el ̂teorema ̂de ̂Stokes(C) ̂el ̂teorema ̂de ̂Green(D) ̂el ̂teorema ̂fundamental ̂del ̂cálculo ̂vectorial

12. ̂¿Por ̂qué ̂son ̂las ̂líneas ̂del ̂campo ̂B ̂no ̂crean ̂o ̂destruyen?̂(A) ̂porque ̂la ̂corriente ̂eficaz ̂siempre ̂cancela ̂la ̂corriente ̂aplicada(B) ̂porque ̂el ̂momento ̂monopolo ̂magnético ̂es ̂siempre ̂el ̂más ̂fuerte(C) ̂porque ̂la ̂energía ̂se ̂conserva(D) ̂porque ̂no ̂hay ̂cargas ̂magnéticas ̂reales

13. ̂¿Por ̂qué ̂tenemos ̂que ̂utilizar ̂la ̂densidad ̂de ̂carga ̂retardada ̂en ̂la ̂ley ̂de ̂Coulomb, ̂cuando ̂la ̂densidad ̂de ̂cargacambios ̂en ̂el ̂tiempo?̂(A) ̂para ̂asegurarse ̂de ̂que ̂la ̂radiación ̂viaja ̂más ̂rápido ̂que ̂c(B) ̂garantizar ̂la ̂causalidad ̂(causa ̂y ̂efecto)(C) ̂para ̂garantizar ̂la ̂conservación ̂de ̂la ̂carga(D) ̂para ̂cancelar ̂el ̂campo ̂infinito ̂a ̂una ̂carga ̂puntual

14. ̂Considere ̂un ̂campo ̂magnético ̂inicialmente ̂estática, ̂pero ̂no ̂uniforme ̂B. ̂¿Cómo ̂podemos ̂hacer ̂que ̂la ̂curvatura ̂de ̂campo ̂E?̂(A) ̂hacer ̂que ̂la ̂fuerza ̂total ̂del ̂campo ̂B ̂oscilan ̂en ̂el ̂tiempo ̂sin ̂cambiar ̂su ̂patrón ̂espacial(B) ̂hacer ̂que ̂el ̂patrón ̂espacial ̂de ̂cambio ̂B ̂continuamente ̂en ̂el ̂tiempo ̂sin ̂cambiar ̂su ̂fuerza ̂global(C) ̂mantener ̂la ̂fuerza ̂total ̂y ̂el ̂patrón ̂de ̂B ̂de ̂la ̂misma, ̂pero ̂cambiar ̂continuamente ̂en ̂el ̂espacio(Todo ̂lo ̂anterior

15. ̂Un ̂gran ̂esfera ̂con ̂una ̂carga ̂electrostática ̂total ̂de ̂+ ̂Q ̂fija ̂permanentemente ̂y ̂de ̂manera ̂uniforme ̂sobre ̂su ̂superficiese ̂pone ̂cerca ̂de ̂un ̂ámbito ̂aún ̂más ̂grande ̂que ̂está ̂llevando ̂a ̂cabo ̂y ̂lleva ̂una ̂carga ̂total ̂+ ̂Q ̂/ ̂2. ̂Ambosesferas ̂están ̂aislados ̂(no ̂conectados ̂a ̂tierra). ̂¿Cuál ̂es ̂la ̂fuerza ̂entre ̂las ̂dos ̂esferas?̂(A) ̂siempre ̂repulsiva(B) ̂siempre ̂atractiva(C) ̂repulsiva ̂cuando ̂están ̂lejos, ̂atractivo ̂cuando ̂están ̂lo ̂suficientemente ̂cerca(D) ̂atractivo ̂cuando ̂están ̂lejos, ̂repulsiva ̂cuando ̂están ̂lo ̂suficientemente ̂cerca