Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite down the exact potential and expand it in the appopiate small paamete to obtain the esult we showed in class, which is that to leading ode Φ( = l cos θ ( Using cosine ule fo the length of R and R shown in Fig.., ( Φ( = + l l cos θ + l + l cos θ ( = ( + (l/ (l/ cos θ + (l/ + (l/ cos θ We now expand this using (l/ as a small paamete. Φ( = ( + (l/ cos θ + (l/ 3 cos θ ( (l/ cos θ + (l/ 3 cos θ l cos θ = + O((l/ 3 (3. Deive the electic field fom E. (. Show that E φ = 0 and wite E and E θ explicity. You ll need the expession of gadient in spheical pola coodinates. E = E ˆ + E θ ˆθ + Eφ ˆφ = Φ = Φ ˆ 4l cos θ = ˆ + 3 Φ θ ˆθ sin θ l sin θ ˆθ + 0 3 Φ φ ˆφ (4
3. Ague that the esult E φ = 0 is exact and is not a esult of the dipole appoximation. The poblem has an axial symmety, so we don t expect the potential to have a φ dependence, exact o othewise. This implies that the E φ component is exactly zeo. 4. Find specific values of θ whee you can ague that the esult of E o of E θ ae exact. You should not need to do any complicated calculations involving the exact potential. Instead, use geneal physical aguments. The tick is to look fo zeos, since thee is often a deepe eason why a uantity is zeo. Fo example, E = 0 in the θ = π/ plane. The deepe eason is that in this x-y plane, the length of R and R ae the same, so we expect Φ = 0 thoughout the plane. This means that any vaiation of the potential in the plane is zeo, so we don t expect any electic field components paallel to the plane, such as E and E φ, to suvive in the plane. We also note that E θ = 0 along the lines θ = 0 and θ = π. The deepe eason is that as we appoach the z-axis, if E θ doesn t tend to zeo, the electic field becomes ill-defined along the axis, because we end up with a situation whee we have electic field lines adiating out of the z-axis as though thee s a line chage. 5. Explain why the elative coection to E due to highe ode tems that we neglected scales like (l/, athe than like (l/ as we might naively expect. As an example, while the lowest ode tem contains (l/, the next lowest ode tem contains (l/ 3 athe than (l/. The most staightfowad answe is to notice that when we expanded the potential, tems in even powes of (l/ cancel out, so fo example, the next highe ode coection to the potential entes at (l/ 3. Theefoe, the atio between adjacent odes in the expansion of Φ, and hence of E, scales as (l/ athe than (l/. Why do tems in even powes of (l/ cancel out? We expect the potential to be antisymmetic unde a eflection about the x-y plane, i.e. when we swap the chages. This means that all tems in the expansion must contain only odd powes of cos θ. A bit of thought should convince you that only odd powes of (l/ can contain odd powes of cos θ. 6. We now move the oigin a bit, that is O new = O old + a, so that thee ae only small
coections to the esults we have obtained so fa. Please define, in wods, what a bit means. (It might help to answe the next pat of the uestion fist. Naively, we might expect that fo the coections to be small, we need a l. Howeve, the esults in the next pat shows that the appoximation actually woks petty well as long as al/ l/, o in othe wods, a/. Theefoe, a bit means that the distance moved should be small compaed to. Fo example, since we know l, choosing an a l is pefectly fine. 7. Now we expand aound the new oigin. Show that to leading ode (that is the fist non-vanishing tem the esult is unchanged. Howeve, unlike pat 5, we now have highe ode coections that can be thought of as being one powe highe in the expansion paamete(s athe than two. Wite them down and show that they vanish when a = 0. Fo convenience, you can define a to be the length of a, θ la the angle between the dipole vecto and a, and θ a the angle between a and. If we now include the shift, the exact expession is given by Φ( = + l + a l cos θ la cos θ la + a cos θ a + l + a + l cos θ + la cos θ la + a cos θ a = + (l/ + (a/ (l/ cos θ (la/ cos θ la + (a/ cos θ a + (l/ + (a/ + (l/ cos θ + (la/ cos θ la + (a/ cos θ a(5 Now expand this in small paametes (l/ and (a/. We get Φ = ( l cos θ + la(cos θ la 3 cos θ a cos θ (6 We notice that to the lowest ode, the esult is unchanged. The next ode is indeed just one powe highe in the expansion paametes, and clealy vanishes when a = 0. 8. Assume that instead of two point-like paticles, thee ae two small blobs with some unknown shapes. The cente of chage of one of them is at +l in the ẑ diection, and the total chage of it is +. The second one is an exact mio image of the fist and its cente of chage is at l in the ẑ diection, and the total chage of it is. Assume that the blobs have typical size b such that b l. Ague that to the appoximation we ae woking in, the esult of E. ( emained unchanged. 3
Since the two blobs ae mio images, this means that fo evey infinitesimal chage element with coodinates (x, y, z, thee is an eual and opposite chage element with coodinates (x, y, z. Each such pai can be thought of as a pai of opposite chages just like the one in the fist pat of this uestion, except that now they ae sepaated by a distance of z, and that the midpoint has been displaced fom the oigin by a distance of x + y b in the x-y plane. Howeve, since we just agued that a small displacement of size up to l can be neglected to the fist appoximation, this means that the pai of chage elements contibute to Φ by an amount dφ = cos θd z = cos θρ(x, y, z dx dy dz z (7 Reaanging the numeato and integating ove the entie blob, we find that Φ = cos θ ρ(x, y, z z dx dy dz = cos θl (8 whee in the last step, we used the definition of the z-component of the cente-of-chage l ρ(x, y, z z dx dy dz. Question : Some algeba fo the multipole expansion In this uestion, you ae asked to show some mathematical esults that ae used in the teatment of the multipole expansion.. Show that ( = 0, fo > 0 (9 and indicate explicitly whee the fact that we use finite is used. Hint: it is athe simple to do in Spheical coodinates. ( = ( We use the fact that > 0 when we set / =. = ( ( = 0 (0. Using summation convention indicate which of the following is a coect expession fo (/? Note thae can be me then one coect answe, you need to indicate all of them. (a δ ij (/ 4
(b x k x k (/ (c (/ (d (/x i (e x k x k (/ x i x i No explenation is needed, just mak the ight ones. The fist, second and last ones ae coect. 3. In class we show that a geneal tem in the expansion can be witten as Φ (l = ( l ( a x l! a,ix a,jx a,k x l a,l. ( x k x l a Show that a simila expession whee two indeces ae the same vanish, that is, show that a δ ij x a,k x l a,l x k x l a ( = 0 ( This esult can then stated that thee ae many ways to define the multipoles by adding things that ae popotional to the zeo above. Using the fact that I do not cae about the ode of taking deivative hee, we see that if we fist apply the i and j and then apply the δ ij on it we see that we have zeo. 5