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Potential inside a grounded spherical conducting shell due to point dipole 
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#1
Feb1009, 10:32 AM

P: 128

1. The problem statement, all variables and given/known data
Suppose a point dipole is located at the centre of a grounded spherical conducting shell. Find the potential and electric field at points inside the shell. (Hint: Use zonal harmonics that are regular at the origin to satisfy the boundary conditions on the shell.) 2. Relevant equations since the metal is grounded then only negative charges are left on the surface. V(r)=[tex]\frac{1}{4\pi\epsilon}[/tex][tex]\sum\frac{q}{r}[/tex] 3. The attempt at a solution i'm not sure the effect of spherical shell on the potential at any points inside the points, anyone can offer some insight thanks? 


#2
Feb1009, 11:06 AM

HW Helper
P: 5,003

You seem to be attempting to use Coulomb's law; but that is a bad idea. The dipole will induce some unknown charge density onto the shell...correct? How can you possibly use Coulomb's law when you don't know what that charge density is?
Instead, use the hint provided. Where does the potential obey Laplace's equation? What is the general solution to Laplace's equation using spherical harmonics? Think of appropriate boundary conditions and apply them to that solution. 


#3
Feb1109, 01:54 AM

P: 128

well, i just informed by professor the point dipole at the origin will have the potential of [tex]\frac{1}{4\pi\epsilon}\frac{p*cos\theta}{r^{2}}[/tex] inside the sphere (p=dipole moment).
the boundary condition at the shell is V=0. and i need 1 more boundary condition to solve the laplace equation, which in this case is the solution is of the form, ([tex]A*_{l}r^{l}+\frac{B_{l}}{r^{l+1}}[/tex]) [tex]P_{l}*cos\theta[/tex]. (the latex has some problem, but the way the theta term in denominator is not meant to be there ad should be shifted outside to be the nominator) just not sure what is the next boundary condition as i cant choose r=0 as the potential fx will blow up. any ideas how to kick start the next step, thanks 


#4
Feb1109, 02:36 AM

Sci Advisor
PF Gold
P: 1,781

Potential inside a grounded spherical conducting shell due to point dipole
I don't know if this helps but consider that since the shell is conducting and grounded the field outside should be zero as should be the potential. Thus the superposition of the fields due to the charge distribution on the sphere and the dipole inside should cancel outside the sphere.
You don't need to solve for the charge distribution on the sphere explicitly you need only determine the component of the potential it must be producing to cancel the potential of the dipole. You can then use the fact that the potentials of both sphere and internal dipole must tend to zero at r>infinity. [Edit: Correction! Not necessarily the potentials but the fields i.e. gradient of the potentials must approach 0 at r>infinity.] 


#5
Feb1109, 07:55 AM

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P: 5,003

[tex]V(r,\theta)=\sum_{l=0}^{\infty}A_l r^l P_l (\cos\theta)[/tex] And so the potential due to the shell and the dipole is of the form: [tex]V(r,\theta)=\frac{1}{4\pi\epsilon_0}\frac{p*\cos\theta}{r^{2}}+\sum_{l= 0}^{\infty}A_l r^l P_l (\cos\theta)[/tex] 


#6
Feb1109, 08:33 AM

P: 128

potential at the origin is zero if no dipole, i supposed. good reasons, but here comes the questions. since the potential is of the function (radius,theta) we need 4 equations (boundary conditions), and what would that be? the only condition i know off is r=R (at the shell) V=0. 


#7
Feb1109, 08:43 AM

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P: 5,003

The general solution only has 2 unknown constants [itex]A_l[/itex] and [itex]B_l[/itex] (Which is the way it should be since it is the solution to a 2nd order differential equation!) The fact that the potential due to the shell is bounded at r=0 allowed you to determine the [itex]B_l[/itex] values. Now just apply the other boundary condition [tex]V(r=R,\theta)=0[/itex] to find the [itex]A_l[/itex] values. 


#8
Feb1109, 08:46 AM

P: 128

[you need only determine the component of the potential it must be producing to cancel the potential of the dipole] how do i determine the potential of the shell itself? assuming i get your meaning. 


#9
Feb1109, 08:52 AM

P: 128




#10
Feb1109, 08:53 AM

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P: 5,003

The solution [tex]V(r,\theta)=\frac{1}{4\pi\epsilon_0}\frac{p*\cos\theta}{r^{2}}+\sum_{l= 0}^{\infty}A_l r^l P_l (\cos\theta)[/tex] is only valid inside the sphere; the boundaries are therefor at r=0 and r=R not r=infinity. In order to find the [itex]A_l[/itex]'s apply the condition at the shell [tex]V(r=R,\theta)=0[/tex]. You still haven't actually used that condition! 


#11
Feb1109, 09:03 AM

P: 128

which means [tex]\sum_{l=0}^{\infty}A_l r^l P_l (\cos\theta)=\frac{1}{4\pi\epsilon_0}\frac{p*\cos\theta}{r^{2}}[/tex] and then multiply both sides by sin(theta)? and integrate it 


#12
Feb1109, 09:06 AM

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P: 5,003

The reason why the [itex]B_1[/itex] term corresponds to the dipole, is because of the form of the potential of the dipole [tex]V_{dip}=\frac{1}{4\pi\epsilon_0}\frac{p*\cos\theta}{r^{2}}=\frac{1}{4\p i\epsilon_0}\frac{p*P_1(\cos\theta)}{r^{1+1}}\implies B_1=\frac{p}{4\pi\epsilon_0}[/tex] 


#13
Feb1109, 09:10 AM

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But a much easier method is to notice that [tex]\cos\theta=P_1(\cos\theta)[/tex] and simply compare coeffecients. 


#14
Feb1109, 09:12 AM

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[tex]\sum_{l=0}^{\infty}A_l R^l P_l (\cos\theta)=\frac{1}{4\pi\epsilon_0}\frac{p*\cos\theta}{R^{2}}[/tex] where capital R is the radius of the shell, right? 


#15
Feb1109, 09:19 AM

P: 128

this is from the [itex]B_l[/itex] term, i got it now. thanks it should be this right? [tex]V_{dip}=\frac{1}{4\pi\epsilon_0}\frac{p*\cos\theta}{r^{2}}=\frac{B_L*p* P_1(\cos\theta)}{r^{1+1}}[/tex] hence comparing it and we get [itex]=B_l=\frac{p}{P_l*4*pi*epsilon}[/itex] 


#16
Feb1109, 09:26 AM

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P: 5,003

What is [tex]\frac{B_1}{r^{1+1}}P_1(\cos\theta)[/tex] if [tex]B_1=\frac{p}{4\pi\epsilon_0}[/tex]? What is [tex]\sum_{l=0}^{\infty}\frac{B_l}{r^{l+1}}P_l(\cos\theta)[/tex] if [tex]B_l=\left\{ \begin{array}{lr} 0, & l\neq1\\ \frac{p}{4\pi\epsilon_0}, & l=1\end{array}[/tex]? How is that not in the form of a zonal harmonic? 


#17
Feb1109, 09:31 AM

P: 128




#18
Feb1109, 09:35 AM

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P: 5,003

When someone writes [tex]P_l(\cos\theta)[/tex], they mean [itex]P_l[/itex] as a function of [itex]\cos\theta[/itex] not [itex]P_l[/itex] times [itex]\cos\theta[/itex]......perhaps that is the source of your confusion? 


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