Potential Energy

Charge and Current Distributions

2.3 Potential Energy

Figure 2.7: Forces on the sides of rectangular current loops in the (a)x-y plane, (b)y-z plane and (c)y-z plane.

we are free to choose the point about which we calculate the torque. For the loop of Figure 2.7, we calculate torques about the lower left corner of the loop in each of the three orientations. So long as the magnetic field intensity,B, is uniform over the area of the loop, there is no net force, hence the torque is independent of origin.

x-y plane, F igure2.7(a) y-z plane, F igure2.7(b) x-z plane, F igure2.7(c) τy = +δxIδyBx= +mzBx τz = +δyIδzBy = +mxBy τz= +δxIδzBx=−myBx

τx=−δyIδxBy =−mzBy τy =−δzIδyBz=−mxBz τx=−δzIδxBz= +myBz

Considering each of these loops as the projection of an arbitrarily oriented loop, we combine these results to obtain generally

τ_x=m_yB_z−m_zB_y τ_y=m_zB_x−m_xB_z τ_z=m_xB_y−m_yB_x

(2–46)

or, more briefly

τ =m ×B (2–47)

A more instructive approach to the potential energy of a charge distribution in an external field is obtained by considering the potential energy,W =#

qiV(r⁽ⁱ⁾) of a collection of point chargesqi. Expanding the potential energy as a Taylor series about the origin, we obtain

q_iV(r⁽ⁱ⁾) =

q_iV(0) +

q_ir⁽ⁱ⁾ ∂V

∂x + 1 2!

q_ir⁽ⁱ⁾r⁽ⁱ⁾_m ∂²V

∂x∂x_m +· · · (2–50) where the superscript (i) on the coordinates indicates the charge to which the coor- dinate belongs. Since the external field cannot have charges at the locationsr⁽ⁱ⁾as these positions are already occupied by the charges subjected to the field, we have

∇ · E =∇²V = 0,which we use to rewriteW as follows:

W =QV +p·∇V +¹₆#

3qix⁽ⁱ⁾ x⁽ⁱ⁾m

∂²V

∂x∂xm

+· · · (2–51)

=QV −p·E +¹₆ q_i

3x⁽ⁱ⁾ x⁽ⁱ⁾_m ∂²V

∂x∂xm −r⁽ⁱ⁾²∇²V

· · · (2–52)

=QV −p·E +¹₆ q_i

3x⁽ⁱ⁾ x⁽ⁱ⁾_m −δ_mr⁽ⁱ⁾² ∂²V

∂x∂x_m· · · (2–53)

=QV −p·E +¹₆Qm

∂²V

∂x∂x_m· · · (2–54)

We assume summation over repeated indices in (2–54).

Example 2.8: Find the energy of the quadrupole of Figure 2.2 in the potential V(x, y) =V₀xy.

Solution: Using the form of the quadrupole energy given in (2-54), we have W = ¹₆

Q_xx∂²V

∂x² +Q_xy ∂²V

∂x ∂y+Q_xz ∂²V

∂x ∂z +Qyx

∂²V

∂y∂x+Qyy

∂²V

∂y² +· · ·

(Ex 2.8.1)

= ¹₆

3qa²V₀+ 3qa²V₀

=qa²V₀ (Ex 2.8.2)

Exercises and Problems

Figure 2.8: The square has alternating posi- tive and negative charges along the edges.

Figure 2.9:Classical model of helium.

2-1 Find the electric dipole moment of a thin ring lying in the x-y plane cen- tered on the origin bearing line charge ρ=λ δ(r−a)δ(z) cosϕ.

2-2 Find the dipole moment of a thin, charged rod bearing charge density ρ= λ z δ(x)δ(y) forz∈(−a, a).

2-3Compute the curl of (2–23) to obtain (2–25).

2-4 Show that the dipole moment of a charge distribution is unique when the monopole (charge) vanishes.

2-5Find the quadrupole moment of two concentric coplanar ring charges q and

−q, having radiia andb respectively.

2-6 Find the quadrupole moment of a square whose edges, taken in turn, have alternating charges ± q uniformly dis- tributed over each as illustrated in Fig- ure 2.8.

2-7Find thegyromagnetic ratio, g(m= gL), for a charged, spinning object whose mass has the same distribution as its charge.

2-8 Find the gyromagnetic ratio of a charged spinning sphere whose mass is uniformly distributed through the vol- ume and whose entire charge is uni- formly distributed on the surface.

2-9 Find the quadrupole moment of a rod of length L bearing charge density ρ = η(z²−L²/12) , with z measured from the midpoint of the rod.

2-10 Show that the potential generated by a cylindrically symmetric quadrupole at the origin is

V = Qzz

16πε₀r³(3 cos²θ−1) 2-11 Find the charge Q contained in a sphere of radiusacentered on the origin, whose charge density varies asρ₀z². 2-12Find the quadrupole moment of the sphere of radiusa of problem 2-11.

2-13 Show that the dipole term of the multipole expansion of the potential can be written

V₂=− 1 4πε₀

qir⁽ⁱ⁾·∇ 1

r

2-14 Show that the quadrupole term of the multipole expansion of the potential can be written

V₃= 1 8πε₀

q_ir⁽ⁱ⁾·∇

r⁽ⁱ⁾·∇

1 r

2-15Use the “multipole expansion” (2–

32) to find the potential due to the rod of (2-2) at points|r|> a.

2-16Use (2–32) to find the potential due to the charged rod of (2-2) at points not on the rod having|r|< a.

2-17 A classical model of the helium atom has two electrons orbiting the nu- cleus. Assuming the electrons have co- planar circular orbits of radiusaand ro- tate with angular frequency ω, find the electric dipole and quadrupole moments as a function of time in each of the fol- lowing cases. (a) The electrons co-rotate diametrically opposed, (b) the electrons counter-rotate (see Figure 2.9).

2-18Find an expression for the force be- tween two dipoles p₁, andp₂, separated byr.

2-19 Find an expression for the force between a quadrupole with components Qmand a dipolepseparated byr.

2-20 If magnetic monopoles existed, their scalar magnetic potential would be given by

V_m(r) = 1 4π

q_m

|r−r|

Obtain the magnetic scalar potential for two such hypothetical monopoles of op- posite sign separated by a small distance a. Show that, if we set qm = m/a, the magnetic scalar potential becomes, in the limit ofa→0, that of a magnetic dipole of strengthqma.

2-21 The proton has a Land´e g-factor of 5.58 (the magnetic moment is m = 2.79e¯h/2m). When a proton is placed in a magnetic induction field, its spin pre- cesses about the field axis. Find the fre- quency of precession.

2-22Obtain an expression for the poten- tial arising from a sheet of dipoles dis- tributed over some surface. Assume a dipole layer densityD =np, where n is the number of dipoles per unit area and p is the mean dipole moment of these dipoles.

2-23 In the Stern-Gerlach experiment, atoms with differently oriented mag- netic moments are separated in pas- sage through a nonhomogeneous mag- netic field produced by a wedge-shaped magnet. Assuming the field has

∂ B

∂z =αˆk

find (classically) the transverse force on atoms whose magnetic moment makes angleθ with thez axis.

2-24Clearly, one could separate electric quadrupoles using an approach similar to that of the Stern-Gerlach experiment.

Given that molecules have quadrupole moments of order 10⁻³⁹ C-m², find the electric field gradient required to impart an impulse of 10⁻²⁶kgm/s to a molecule travelling at 100 m/s through a 1 mm region containing the gradient.

Chapter 3