Universidad Nacional de ColombiaElectrodinamica - II (II-2015)Ejercicios: Lista 1

Profesor Carlos ViviescasFecha de entrega: 20.8.15

Exercise 1 (4 points) Penning trap

Consider an electron (charge −e) moving in a region where a uniform magnetic field B = Bez overlapswith a quadrupolar electric potential

Φ(r) =U0

2r20(x2 + y2 − 2z2) ,

where U0 is a fix value of the potential and r0 is the confinement radius.

a) Asume first that U0 = 0. What kind of trajectories follows the electron on the x-y plane? Deter-mine for this case the frequency of movement.

b) Take now U0 6= 0. Show that along the z-axis the electron follows an oscillatory movement withfrequency ωz. Determine ωz.

c) Write the complete equations of motion in the x-y plane and show that there exist circular tra-jectories that are particular solution of these equations of motion. Evaluate the correspondingfrequencies (known as magneton frequencies) ωM and express them in terms of the frequenciesfound in items a) and b).

Exercise 2 (4 points) The Levi–Civita tensor

The Levi–Civita tensor is defined by

εijk =

ε123 = ε231 = ε312 = +1 ,

ε132 = ε213 = ε321 = −1 ,

εijk = 0 if there is a repeated index.

1. Show that

εijkεilm = δjlδkm − δjmδkl .

2. Use εijk and the repeated indices convention to write the vectorial product a× b in terms of thecomponents of the vectors a and b.

3. Evaluate with the help of the Levi–Civita tensor

(a) ∇× (∇φ) = rot (gradφ)what is its physical meaning?

(b) ∇× (∇×A) = rot (rotA)

Exercise 3 (4 points) Charge and currents distributions

Using the Dirac-delta function in the proper system of coordinates, write down the following chargedistributions ρ(r) and current distributions j(r) as distributions in three dimensions:

1. The charge density for a total charge Q uniformly distributed on the surface of a sphere of radiusR using spherical coordinates.

2. Assume now that the sphere in the previous item spins around the z-axis, which crosses it alongits center, with an angular velocity ω = ωez. Find the associated current density in sphericalcoordinates.

3. The charge density for a charge λ per unit of length uniformly distributed on a cylindrical surfaceof radius b. Use cylindrical coordinates.

4. The charge density for a charge Q that uniformly covers a circular disk of negligible thick andradius R. Both in cylindrical and spherical coordinates.

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5. Assume now that the disk in the previous item spins around its center with an angular velocityω. Find the current density in cylindrical and spherical coordinates.

6. The current density in a circular loop of radius a carryng a current I. Use cylindrical coordinates.

Hint: Use your physical intuition to write down the distributions.

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