Static Electric and Magnetic Fields in Vacuum
1.2 Moving Charges
1.2.6 The Magnetic Scalar Potential
The vector potential of a uniform magnetic induction field may be written as A( r) = B ×r
2 (1–56)
which is easily verified as follows:
∇ × A =∇ ×
B ×r 2
=B(∇ · r)−r(∇ · B) + (r·∇ )B −(B ·∇) r 2
= 3B −B
2 =B (1–57)
Figure 1.15: The area enclosed by the loop points in the direction shown, determined by the direction of the current.
enclosed is immaterial; only the boundary is significant. If the observer moves by an amount dr or, equivalently, the loop moves by−dr, the solid angle subtended by the loop will change. In particular, the area gained by any segment d of the loop is−dr×d, giving a change in solid angle subtended (see Figure 1.16):
dΩ = [(−dr)× d]·(−R) R3
= dr·(d ×R)
R3 (1–62)
ComparingdΩ, the integrand of (1–62) withdVm, (1–60), we find that dVm=−I
4πdΩ (1–63)
Since we may in general writedVm=∇Vm·dranddΩ =∇Ω·dr, we conclude that
Vm=−I
4πΩ (1–64)
Figure 1.16: Although the area does not change under displacement, the solid angle does, because the meanRchanges.
serves as a scalar potential for the magnetic induction field.
Example 1.15: Find the magnetic scalar potential at a point below the center of a circular current loop of radiusa(Figure 1.17). Use the scalar potential to find the magnetic induction field on the central axis.
Figure 1.17:The observer is directly below the center of the loop.
Solution: We first find Ω. To this end, we noteR =zˆk−rr,ˆ R=
z2+r2 and R·d A=zrdrdϕ, so that
−Ω =
d A·R R3 =
a 0
2π
0
zrdrdϕ (z2+r2)3/2
= −2πz
√z2+r2 a
0
= 2π
1− z
√z2+a2
(Ex 1.15.1) from which we conclude
Vm=−I 2
z
√z2+a2 −1
(Ex 1.15.2) (the constant term may of course be dropped without loss). As shown in the next example, (1.16), the scalar potential of the current loop becomes a building block for a number of other problems whose currents can be decomposed into current loops.
The magnetic induction field along the z axis previously obtained in example 1.8 is now easily found:
Bz(0,0, z) =−µ0 ∂
∂zVm
= Iµ0 2
1
(z2+a2)1/2− z2 (z2+a2)3/2
= Iµ0 2
a2
(z2+a2)3/2 (Ex 1.15.3)
Example 1.16: Find the scalar magnetic potential along the axis of a solenoid of lengthLwithN closely spaced turns, carrying currentI. Assume the pitch may be neglected.
Solution: We choose our coordinate system so that thez axis runs along the axis of the solenoid and the origin is at the center of the solenoid. Using the preceding example 1.15, the magnetic scalar potential of a coil at the origin is
Vm=−I 2
√ z
z2+a2 (Ex 1.16.1)
If the loop were located at position z instead of the origin, the expression for Vmwould become
Vm(z) =−I 2
(z−z)
(z−z)2+a2 (Ex 1.16.2) The scalar potential fromN turns at varying locationsz is then obtained by sum- ming the potential from each of the loops:
Vm(z) =−I 2
(z−z)dN
(z−z)2+a2 =−N I 2L
L/2
−L/2
(z−z)dz (z−z)2+a2
= N I 2L
(z−z)2+a2 L/2
−L/2
=−N I 2L
z+12L2
+a2−
z−12L2 +a2
(Ex 1.16.3) The calculation of the magnetic induction field along the axis of the solenoid is now a simple matter. (Exercise 1-23)
Exercises and Problems
Figure 1.18:A Helmholtz coil consists of two identical parallel coils, spaced at their common radius.
Figure 1.19:Geometry of the solenoid of ques- tion 21.
1-1Verify that Ex 1.3.4 yields the limit- ing formQ/4πε0z2 whenz→0.
1-2Find the electric field along the axis of a charged ring of radiusalying in the x-yplane when the charge density on the ring varies sinusoidally around the ring as
ρ=λ0(1 + sinϕ)δ(r−a)δ(z) 1-3 Find the electric field above the center of a flat circular plate of radius a when the charge distribution is ρ = br2δ(z) whenr ≤a and 0 elsewhere.
1-4 Find the electrostatic field estab- lished by two long concentric cylinders having radii a andb, bearing respective surface charge densitiesσaandσb, in the region inside the inner cylinder, between the cylinders, and outside the two cylin- ders.
1-5Find the electrostatic field produced by a spherical charge distribution with charge densityρ0e−kr.
1-6 Find the electric field produced by a spherically symmetric charge distribu- tion with charge density
ρ=
ρ0
1−r a
2
forr≤aand
0 forr≥a
1-7 Two large parallel flat plates bear uniform surface charge densities σ and
−σ. Find the force on one of the plates due to the other. Neglect the fringing fields. Note that just using the electric field between the plates to calculate the force asσ EAgives twice the correct re- sult. (Why?)
1-8 Find the electric field at any point (not on the charge) due to a line charge with charge density
ρ=
bzδ(x)δ(y) z ∈ (−a, a)
0 elsewhere
lying along thezaxis betweenaand−a.
(Fairly messy integral!)
1-9 Within a conductor, charges move freely until the remaining electric field is zero. Show that this implies that the electric field near the surface of a conduc- tor is perpendicular to the conductor.
1-10 Find the electric potential for the charge distribution of problem 1-8.
1-11 Find the electric potential for the charge distribution of problem 1-2.
1-12 Find the electric potential for the charge distribution of problem 1-3.
1-13 A fine needle emits electrons iso- tropically at a steady rate. Find the di- vergence of the current density and the resulting current flux at distance r from the point in the steady state.
1-14 Using the magnetic scalar poten- tial, find the magnetic induction field along the spin axis of a uniformly char- ged spinning disk.
1-15 Using the results from 1-14, find the magnetic induction field along the spin axis of a uniformly charged spinning thin spherical shell of radiusRat a point outside the shell.
1-16 Using the results from 1-15, find the magnetic induction field along the spin axis of a uniformly charged spinning sphere of radiusRat a point outside the sphere.
1-17 A Helmholtz coil consists of two parallel plane coils each of radiusa and spaced bya(Figure 1.18). Find the mag- netic induction field at the center (a/2 from the plane of each coil).
1-18Use the Biot-Savart law to find the axial magnetic induction field of a closely wound solenoid having length L and N closely spaced turns. (Ignore the pitch of the windings.)
1-19Find the magnetic induction field of a long coaxial cable carrying a uniformly distributed current on its inner conduc- tor (radius a) and an equal counter- current along its outer conductor of ra- diusb and thickness<<b.
1-20Expresses the vector potential of a filamentary current loop as a line inte- gral and show that the magnetic vector potential of the current loop vanishes.
1-21Show that the magnetic vector po- tential satisfies∇2A=−µ0J.
1-22 Find the vector potential of two long parallel wires each of radius a, spaced a distancehapart, carrying equal currentsI in opposing directions.
1-23 A low-velocity beam of charged particles will spread out because of mu- tual repulsion. By what factor is the mu- tual repulsion reduced when the particles are accelerated to .99c?
1-24 Differentiate the axial scalar mag- netic potential of a solenoid of lengthL with N closely spaced turns to obtain the axial magnetic induction field. Show that this field can be conveniently writ- ten
Bz= N Iµ0
2L (cosθ1+ cosθ2) whereθ1 andθ2are respectively the an- gles subtended by the solenoid’s radius at the left and the right end of the coil (Figure 1.19).
1-25A charged particle in a crossed elec- tric and magnetic induction field experi- ences no net force when it moves through the fields with an appropriate velocity known as theplasma drift velocity. Find an expression for the plasma drift veloc- ity.