Energy and Momentum
4.5 Duality Transformations
We might see how Maxwell’s equations would be altered to accommodate magnetic monopoles. Clearly we can no longer sustain∇ · B = 0, but have instead∇ · B = µ0ρm. Moving monopoles would constitute a magnetic current and presumably add a term to the∇ × E equation. Postulating
∇ × E =−∂ B
∂t −µ0Jm (4–77)
we verify a continuity equation for magnetic monopoles:
0 =∇ · (∇ × E) = −∂
∂t(∇ · B)−µ0∇ · Jm (4–78) or
∂ρm
∂t +∇ · Jm= 0 (4–79)
The generalized Maxwell’s equations then become
∇ · E =ρe
ε0 ∇ · B =µ0ρm
∇ × E =−∂ B
∂t −µ0Jm ∇ × B = 1 c2
∂ E
∂t +µ0Je
(4–80) The generalized equations present considerable symmetry. A duality transfor- mation defined by
E=E cosθ+c Bsinθ c B=−Esinθ+c Bcosθ
(4–81) together with
cρe=cρecosθ+ρmsinθ ρm=−cρesinθ+ρmcosθ (4–82) and the associated current transformations
cJe =cJecosθ+Jmsinθ Jm =−cJesinθ+Jmcosθ (4–83) leaves Maxwell’s equations and therefore all ensuing physics invariant. Thus with the appropriate choice of mixing angle θ, magnetic monopoles may be made to appear or disappear at will. So long as all charges have the same ratioρe/ρm (the same mixing angle), the existence or nonexistence of monopoles is merely a matter of convention. The Dirac monopole, on the other hand, would have a different mixing angle than customary charges.
Exercises and Problems
Figure 4.6: The concentric cylinders of prob- lem 4-15.
Figure 4.7:The variable capacitor of problem 4-16.
4-1 Find the potential energy of eight equal charges q, one placed at each of the corners of a cube of sidea.
4-2 Find the inductance of a closely wound solenoid of radius R and length L havingN turns whenRL.
4-3 Find the inductance of a closely wound toroidal coil of N turns with mean radiusband cross-sectional radius a using energy methods.
4-4Find the inductance of a coaxial wire whose inner conductor has a radiusaand whose outer conductor has inner radiusb and the same cross-sectional area as the inner conductor. Assume the same cur- rent runs in opposite directions along the inner and outer conductor. (Hint: The magnetic induction field vanishes outside the outer conductor, meaning that the volume integral ofB2 is readily found.) 4-5 Find the magnetic flux Φ enclosed by a rectangular loop of dimensions× (h−2a) placed between the two conduc- tors of figure 4.2. Compare the induc- tance computed as L = dΦ/dI to the result of example 4.4.
4-6Find the energy of a chargeQspread uniformly throughout the volume of a sphere of radiusa.
4-7Find the energy of a spherical charge
whose density varies as ρ=ρ0
1−r2
a2
forr≤aand vanishes whenr exceedsa.
4-8Find the electric field strength in a light beam emitted by a 5-watt laser if the beam has a 0.5 mm2 (assume uni- form) cross section.
4-9Show that the surface integral of (4–
43) vanishes for the parallel wire field at sufficiently large distance. Chose as vol- ume of integration a large cylinder cen- tered on one of the wires.
4-10Estimate the mutual inductance of two parallel circular loops of radius a spaced by a small distanceb (b a).
4-11 A superconducting solenoid of length L and radius a carries current I in each of its N windings. Find the ra- dial force on the windings and hence, the tensile strength required of the wind- ings. (Note that the radial force result- ing from the interior B is outward di- rected whereas the force on the end faces is inward.)
4-12 Find the force between two prim- itive magnetic monopoles, and compare this force to the force between two elec- tric chargese.
Figure 4.8: A directional Power meter may constructed by attaching the field coils of a power meter to an inductive loop and the movement coil to a capacitive element inside a coaxial conductor.
4-13Show that the generalized Maxwell equations (4-44) are invariant under a duality transformation.
4-14 Find approximately the mutual capacitance of two 1-m radius spheres whose centers are separated by 10 m.
4-15 A pair of concentric cylinders of radiiaandb, (Figure 4.6) are connected to the terminals of a battery supplying an EMF V. Find the force in the axial direction on the inner cylinder when it is partially extracted from the outer one.
4-16 A variable capacitor has 15 semi- circular blades of radiusRspaced at dis- tance d (Figure 4.7). Alternate blades are charged to ±V. The capacitance is varied by rotating one set of blades about an axis on the center of the diameter of the blades with respect to the other set.
Find the electrostatic torque on the mov- able blades when partly engaged. Ne- glect any fringing fields. Could one rea- sonably design an electrostatic voltmeter using this principle?
4-17A power meter may be constructed by using a signal proportional to the cur- rent to power the field (instead of the
customary permanent magnet) of a gal- vanometer and a signal proportional to the voltage to power the moving coil of the galvanometer. A directional power meter may be constructed as illustrated in figure 4.8. Given that the torque on the needle isτ = kI1I2 where I1 is the current induced in the loop by the os- cillating current in the coaxial wire and I2 is the current required to capacitively charge the plate adjacent to the cental conductor, write an expression for the torque on the needle in terms of fre- quency, loop area and plate area, curved to maintain constant distance from the central conductor and the distance of each from the center.
4-18 A spherical soap bubble has a chargeQ distributed over its surface. If the interior pressure and exterior pres- sure are the same, find the radius at which the compressive force from the surface tension balances the repulsive electrical force. Is the radius stable against perturbations? (Hint: the sur- face tension T on a spherical surface with radius of curvature r gives an in- ward pressure of 2T/r.)