Electric potential for an axial quadrupole

In summary, to find the electric potential for an axial quadrupole with point charges q, -2q, and q over the z axis at distances l, 0, and l from the origin, we can use the formula given by Wikipedia for an axial multipole. This formula involves zonal harmonics, which are functions of the form φn=rnPn(cosθ) or φn=r-(n+1)Pn(cosθ) where Pn(cosθ) are Legendre polynomials. To apply this formula to the problem at hand, we first find the potential using superposition of the three point charges, and then expand it in a Taylor series for r>>l. This will result in a functional dependence that
  • #1
Telemachus
835
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Find the electric potential for an axial quadrupole: point charges q, -2q, q over the z axis at distances l,0,l from the origin. Find the electric potential only for distances r>>l and demonstrate that the potential is proportional to one of the zonal armonics.

Well, I found at wikipedia that an axial multipole has an electric potential given by:
[tex]\Phi(r)=\frac{1}{4\pi \epsilon_0 r}\sum_{k=0}^{\infty}qa^k \left ( \frac{1}{r^{k+1}} \right ) P_k(\cos\theta)[/tex]

But I don't know how to apply this to my problem. I don't know neither what the zonal armonics are.
 
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  • #2
Zonal harmonics are functions of the form

φn=rnPn(cosθ) or φn=r-(n+1)Pn(cosθ) where Pn(cosθ) are Legendre polynomials.

In relation to your problem, first find the potential using simple superposition of three point charges, then expand in Taylor series for r>>l. You should get something that matches the functional dependence of one of the two forms. Can you predict which one?
 
  • #3
Thanks.
 

Related to Electric potential for an axial quadrupole

1. What is an axial quadrupole in terms of electric potential?

An axial quadrupole is a type of electric field configuration where the electric potential varies along the axis of symmetry. This means that the electric potential is higher at one end of the axis and lower at the other end.

2. How is the electric potential calculated for an axial quadrupole?

The electric potential for an axial quadrupole can be calculated using the formula V = kQz/r^2, where V is the electric potential, k is the Coulomb's constant, Q is the charge, z is the distance along the axis of symmetry, and r is the distance from the axis of symmetry.

3. What is the significance of electric potential in an axial quadrupole?

The electric potential in an axial quadrupole determines the strength and direction of the electric field. It also affects the motion and behavior of charged particles within the field.

4. How does the electric potential change as the distance from the axis of symmetry increases in an axial quadrupole?

In an axial quadrupole, the electric potential decreases as the distance from the axis of symmetry increases. This is because the electric potential is inversely proportional to the square of the distance from the axis.

5. What are some real-life applications of electric potential in an axial quadrupole?

Axial quadrupoles are commonly used in particle accelerators, mass spectrometers, and ion traps. They are also used in medical imaging technologies such as MRI machines, where the electric potential is used to create a magnetic field to produce images of the human body.

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