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Problem 5. Electric quadrupole Four charged particles are located in the x-y plane each a distance d from the origin. The two charge particles on the x axis have charge q. The two on the y axis have charge -q. :Y a. Determine the electric field at (x,y,z) = (0,0,0). b. Determine the electric field at any point on the z axis, i.e. at (0,0, z). c. Find an exact expression for the electric field along the x axis, Ē(x,0,0), for x > d, -d < x < d and x < -d. d. Find an approximate expression for your result in part c in the limit x >> d. You will need to evaluate all four terms to the same order in d/x. That means that for two of the terms, you will have to carry out the binomial approximation to second order, namely a(a – 1), (1+ 8)ª - 1+ ad + 2 You will obtain a quadrupole field that falls off with distance as 1/distance". Hopefully you are, by now, seeing pattern. In your next E&M class you will explore what is referred to as a "multipole" expansion of electric fields which involves a sum of terms that fall off with distance, r, as 1/r2+n and that involve multipole moments that reflect the geometry of the charge distribution. The monopole moment is the total charge. The dipole moment you explored in problem 2. The quadrupole moment and higher moments are more complicated. But observe that you can write the quadrupole field in terms of a product of q and an area. Also look back at the correction term in part d of problem 4. It is also a quadrupole term.

Problem 5. Electric quadrupole Four charged particles are located in the x-y plane each a distance d from the origin. The two charge particles on the x axis have charge q. The two on the y axis have charge -q. :Y a. Determine the electric field at (x,y,z) = (0,0,0). b. Determine the electric field at any point on the z axis, i.e. at (0,0, z). c. Find an exact expression for the electric field along the x axis, Ē(x,0,0), for x > d, -d < x < d and x < -d. d. Find an approximate expression for your result in part c in the limit x >> d. You will need to evaluate all four terms to the same order in d/x. That means that for two of the terms, you will have to carry out the binomial approximation to second order, namely a(a – 1), (1+ 8)ª - 1+ ad + 2 You will obtain a quadrupole field that falls off with distance as 1/distance". Hopefully you are, by now, seeing pattern. In your next E&M class you will explore what is referred to as a "multipole" expansion of electric fields which involves a sum of terms that fall off with distance, r, as 1/r2+n and that involve multipole moments that reflect the geometry of the charge distribution. The monopole moment is the total charge. The dipole moment you explored in problem 2. The quadrupole moment and higher moments are more complicated. But observe that you can write the quadrupole field in terms of a product of q and an area. Also look back at the correction term in part d of problem 4. It is also a quadrupole term.

Physics for Scientists and Engineers: Foundations and Connections
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Katz, Debora M.
Chapter24: Electric Fields
Section: Chapter Questions
Problem 79PQ
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Problem 5. Electric quadrupole Four charged particles are located in the x-y plane each a distance d from the origin. The two charge particles on the x axis have charge q. The two on the y axis have charge -q. :Y b. a. Determine the electric field at (x,y,z) = (0,0,0). b. Determine the electric field at any point on the z axis, i.e. at (0,0, z). c. Find an exact expression for the electric field along the x axis, E(x,0,0), for x > d, -d < < d and x < -d. d. Find an approximate expression for your result in part c in the limit x >> d. You will need to evaluate all four terms to the same order in d/x. That means that for two of the terms, you will have to carry out the binomial approximation to second order, namely а(а — 1) (1+ 8)ª 2 1+ að + You will obtain a quadrupole field that falls off with distance as 1/distance“. Hopefully you are, by now, seeing pattern. In your next E&M class you will explore what is referred to as a "multipole" expansion of electric fields which involves a sum of terms that fall off with distance, r, as 1/r2+n and that involve multipole moments that reflect the geometry of the charge distribution. The monopole moment is the total charge. The dipole moment you explored in problem 2. The quadrupole moment and higher moments are more complicated. But observe that you can write the quadrupole field in terms of a product of q and an area. Also look back at the correction term in part d of problem 4. It is also a quadrupole term.
Transcribed Image Text:Problem 5. Electric quadrupole Four charged particles are located in the x-y plane each a distance d from the origin. The two charge particles on the x axis have charge q. The two on the y axis have charge -q. :Y b. a. Determine the electric field at (x,y,z) = (0,0,0). b. Determine the electric field at any point on the z axis, i.e. at (0,0, z). c. Find an exact expression for the electric field along the x axis, E(x,0,0), for x > d, -d < < d and x < -d. d. Find an approximate expression for your result in part c in the limit x >> d. You will need to evaluate all four terms to the same order in d/x. That means that for two of the terms, you will have to carry out the binomial approximation to second order, namely а(а — 1) (1+ 8)ª 2 1+ að + You will obtain a quadrupole field that falls off with distance as 1/distance“. Hopefully you are, by now, seeing pattern. In your next E&M class you will explore what is referred to as a "multipole" expansion of electric fields which involves a sum of terms that fall off with distance, r, as 1/r2+n and that involve multipole moments that reflect the geometry of the charge distribution. The monopole moment is the total charge. The dipole moment you explored in problem 2. The quadrupole moment and higher moments are more complicated. But observe that you can write the quadrupole field in terms of a product of q and an area. Also look back at the correction term in part d of problem 4. It is also a quadrupole term.
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