The Axion Electromagnetic Response of Topological Insulators

Home , Anyon

The axion electromagnetic response of topological insulators

Jeroen van den Brink

Flavio Nogueira Zohar Nussinov

Colloquium Wuerzburg 08.7.2019 Marriage of Josephson and Witten effect

• JosephsonJosephson effect effect: current due to phase difference

Superconductor 2 = 1 2 1 IJ = Ic sin() 2 Superconductor 1 @t =2eV Marriage of Josephson and Witten effect

• JosephsonJosephson effect effect: current due to phase difference

Superconductor 2 = 1 2 1 IJ = Ic sin() 2 Superconductor 1 @t =2eV

Witten effect charge fractionalisation due to magnetic monopoles ✓ Q = e n 2⇡ ✓ ◆ e2✓ Axion = 2 E B L 4⇡ · Monopole 3D topological insulator

3D TI: insulating in bulk, conducting Dirac cone on surface

How do Dirac electrons couple to electromagnetic field? 3D topological insulator

3D TI: insulating in bulk, conducting Dirac cone on surface

How do Dirac electrons couple to electromagnetic field?

More precisely: what is the effective EM Lagrangian after integrating out electrons? L 3D topological insulator

3D TI: insulating in bulk, conducting Dirac cone on surface

How do Dirac electrons couple to electromagnetic field?

More precisely: what is the effective EM Lagrangian after integrating out electrons? L 1 Maxwell: = (E2 B2) ⇢ j A LM 8⇡ · 3D topological insulator

3D TI: insulating in bulk, conducting Dirac cone on surface

How do Dirac electrons couple to electromagnetic field?

More precisely: what is the effective EM Lagrangian after integrating out electrons? L 1 Maxwell: = (E2 B2) ⇢ j A LM 8⇡ · 1 e2✓ Answer: = (E2 B2)+ E B ⇢ j A L 8⇡ 4⇡2 · ·

Axion or ✓ term Origin of ✓ -term

✓ is given by the momentum space Chern-Simons form 1 2 ✓ = d3k ✏ijk Tr @ i 4⇡ Ai jAk 3AiAjAk Z  with the non-Abelian ↵(k)= i u↵ @ u Berry vector potential Ai h k| i| ki @ Qi, Hughes & Zhang, and @i = PRB 78, 195424 (2008) @ki Origin of ✓ -term

trivial insulator 3D TI Consequence of ✓ -term: Witten effect work out axion electrodynamics with Lagrangian… @ 1 e2✓ D = L = (E2 B2)+ E B ⇢ j A @E L 8⇡ 4⇡2 · · @ H = L @B

D = ⇢ B = ⇢ r · r · m

Witten, Phys. Lett. B 86, 283 (1979) Consequence of ✓ -term: Witten effect work out axion electrodynamics with Lagrangian… @ 1 e2✓ D = L = (E2 B2)+ E B ⇢ j A @E L 8⇡ 4⇡2 · · @ H = L allow for magnetic monopoles… @B e2✓ D = ⇢ B = ⇢m E =4⇡ ⇢ ⇢ r · r · !r· 4⇡2 m ✓ ◆

ne 2⇡/e Witten, Phys. Lett. B 86, 283 (1979) Consequence of ✓ -term: Witten effect work out axion electrodynamics with Lagrangian… @ 1 e2✓ D = L = (E2 B2)+ E B ⇢ j A @E L 8⇡ 4⇡2 · · @ H = L allow for magnetic monopoles… @B e2✓ D = ⇢ B = ⇢m E =4⇡ ⇢ ⇢ r · r · !r· 4⇡2 m ✓ ◆ … so that the total charge Q is e2✓ ✓ Magnetic monopole Q = q e = e n 4⇡2 m 2⇡ fractionalizes electric ✓ ◆ charge !!

ne 2⇡/e Witten, Phys. Lett. B 86, 283 (1979) Uniform ✓ in this case the axion term in the action is a total derivative

for an infinite 3D TI, Maxwell equations do not modify Uniform ✓ in this case the axion term in the action is a total derivative

for an infinite 3D TI, Maxwell equations do not modify

but on the surface of a TI ✓ changes! zˆ ✓ @✓/@z

TI Non-Uniform ✓ e2 Gauss’ law E =4⇡⇢ ✓ B r · ⇡ r ·

zˆ ✓ @✓/@z

TI Non-Uniform ✓ e2 Gauss’ law E =4⇡⇢ ✓ B r · ⇡ r · @E e2 @✓ Ampere’s law B =4⇡ + j + ✓ E + B r⇥ @t ⇡ r ⇥ @t ✓ ◆ ✓ ◆

zˆ ✓ @✓/@z

TI Non-uniform : induced magnetic monopole REPORTS ✓

that such a general definition of a topological insulator reduces to the Z2 topological insulators Qi, Li, Zang & Zhang, described in (4–6) for non-interacting band Inducing a Magnetic Monopole with insulators; this finding is a three-dimensional (3D) generalization of the quantum spin Hall Science 323, 1148 (2009) insulator in two dimensions (7–10). For generic Topological Surface States band insulators, the parameter q has a micro- scopic expression of the momentum space Xiao-Liang Qi,1 Rundong Li,1 Jiadong Zang,2 Shou-Cheng Zhang1 * Chern-Simons form (3, 11). Recently, experimental evidence of the topologically nontrivial Existence of the magnetic monopole is compatible with the fundamental laws of nature; surface states has been observed in Bi Sb alloy however, this elusive particle has yet to be detected experimentally. We show theoretically that an 1−x x (12), which supports the theoretical prediction electric charge near a topological surface state induces an image magnetic monopole charge due that Bi Sb is a Z topological insulator (4). to the topological magneto-electric effect. The magnetic field generated by the image magnetic 1−x x 2 With periodic temporal and spatial boundary monopole may be experimentally measured, and the inverse square law of the field dependence conditions, the partition function is periodic in q can be determined quantitatively. We propose that this effect can be used to experimentally realize under the 2p shift, and the system is invariant a gas of quantum particles carrying fractional statistics, consisting of the bound states of the under the time-reversal symmetry at q = 0 and electric charge and the image magnetic monopole charge. q = p.However,withopenboundaryconditions, the partition function is no longer periodic in q, he electromagnetic response of a conven- Furthermore, when a periodic boundary condi- and time-reversal symmetry is generally broken tional insulator is described by a dielectric tion is imposed in both the spatial and temporal (but only on the boundary), even when q =(2n + Tconstant e and a magnetic permeability m. directions, the integral of such a total derivative 1)p.Ourworkin(3)givesthefollowingphysical An electric field induces an electric polarization, term is always quantized to be an integer; i.e., interpretation: Time-reversal invariant topologi- S q cal insulators have a bulk energy gap but have whereas a magnetic field induces a magnetic ℏ qn (where n is an integer). Therefore, the polarization. As both the electric field E(x)and partition¼ function and all physically measurable gapless excitations with an odd number of Dirac the magnetic induction B(x)arewelldefinedinside quantities are invariant when the q parameter is cones on the surface. When the surface is coated an insulator, the linear response of a convention- shifted by 2p times an integer (2). Under time- with a thin magnetic film, time-reversal sym- al insulator can be fully described by the effective reversal symmetry, eiqn is transformed into e–iqn metry is broken, and an energy gap also opens up 2 1 3 2 1 2 3 (here, i = –1). Therefore, all time-reversal in- at the surface. In this case, the low-energy theory action S0 8p ∫d xdt eE − m B ,whered xdt is ¼ variant insulators fall into two general classes, is completely determined by the surface term in the volume element of space and time. However, described by either q =0orq = p (3). These two Eq. 1. As the surface term is a Chern-Simons in general, another possible term is allowed in the time-reversal invariant classes are disconnected, term, it describes the quantum Hall effect on effective action, which is quadratic in the elec- and they can only be connected continuously by the surface. From the general Chern-Simons- tromagnetic field, contains the same number of time-reversal breaking perturbations. This classi- Landau-Ginzburg theory of the quantum Hall derivatives of the electromagnetic potential, and fication of time-reversal invariant insulators in effect (13), we know that the coefficient q = is rotationally invariant; this term is given by terms of the two possible values of the q (2n +1)p gives a quantized Hall conductance of q a 3 e2 1 e2 Sq ∫d xdtE⋅B.Here,a (where ħ parameter is generally valid for insulators with sxy n . This quantized Hall effect on ¼ 2p 2p ¼ ℏc ¼ þ 2 h is PlanckÀÁ’sconstantÀÁ h divided by 2p and c is the arbitrary interactions (3). The effective action the surfaceÀÁ is the physical origin behind the speed of light) is the fine-structure constant, and q contains the complete description of the topological magneto-electric (TME) effect. can be viewed as a phenomenological parameter electromagnetic response of topological insula- Under an applied electric field, a quantized in the sense of the effective Landau-Ginzburg tors. Topological insulators have an energy gap in Hall current is induced on the surface, which in theory. This term describes the magneto-electric the bulk, but gapless surface states protected by turn generates a magnetic polarization and vice effect (1), where an electric field can induce a the time-reversal symmetry. We have shown (3) versa. magnetic polarization, and a magnetic field can induce an electric polarization. Unlike conventional terms in the Landau- Fig. 1. Illustration of the image charge Ginzburg effective actions, the integrand in Sq and monopole of a point-like electric is a total derivative term, when E(x) and B(x) charge. The lower-half space is occupied are expressed in terms of the electromagnetic by a topological insulator (TI) with di- vector potential (where ∂m denotes the partial electric constant e2 and magnetic perme- derivative; m, n, r, and t denote the spacetime ability m2.Theupper-halfspaceisoccupied coordinates; F mn is the electromagnetic field by a topologically trivial insulator (or vac- tensor; and Am is the electromagnetic potential) uum) with dielectric constant e1 and magnetic permeability m1. A point electric q a 3 mn rt charge q is located at (0, 0, d). When seen Sq ∫d xdtemnrtF F from the lower-half space, the image ¼ 2p 16p (1) q a 3 m n r t electric charge q1 and magnetic monopole ∫d xdt∂ emnrsA ∂ A ¼ 2p 4p ð Þ g1 are at (0, 0, d); when seen from the upper-half space, the image electric charge q2 and magnetic monopole g2 1Department of Physics, Stanford University, Stanford, CA are at (0, 0, −d). The red solid lines 94305–4045, USA. 2Department of Physics, Fudan Uni- represent the electric field lines, and blue versity, Shanghai, 200433, China. solid lines represent magnetic field lines. *To whom correspondence should be addressed. E-mail: (Inset) Top-down view showing the in-plane component of the electric field at the surface (red arrows) [email protected] and the circulating surface current (black circles).

1184 27 FEBRUARY 2009 VOL 323 SCIENCE www.sciencemag.org REPORTS

We propose a manifestation of the TME ef- (0, 0, −d); the magnetic field is given by an image by the in-plane component of the electric field fect. When a charged particle is brought close to magnetic monopole g2 at (0, 0, −d). The above and is perpendicular to this component. This the surface of a topological insulator, a magnetic ansatz satisfies the Maxwell equations on each current is nothing but the quantized Hall current monopole charge is induced as a mirror image of side of the boundary. At the boundary z =0,the mentioned earlier. For the problem under consider- the electric charge. The full set of electromag- solution is then matched according to the standard ation, the surface current density is calculated as netic field equations can be obtained from the boundary condition, giving e2 q r q q ^ functional variation of the action S0 + Sq (14), and 1 2 j P3 2 2 3 eϕ 5 ¼ ¼ h 1 a P 2 2 2 ð Þ they can be presented as conventional Maxwell’s 1 e − e 1=m 1=m − 4a2P2 3 r d 1 2 1 2 3 q þ ð þ Þ equations but with the modified constituent equa- ð Þð þ Þ 2 2 which is circulating around the origin (inset of ¼ e1 e1 e2 1=m1 1=m2 4a P3 tions describing the TME effect (3) ð þ Þð þ Þþ Fig. 1) (here, r is the radial distance and^eϕ is the g1 −g2 tangental unit vector). Physically, this surface D E 4pP − 2aP3B ¼ ¼ þ 2 4aP3 current is the source that induces the magnetic H B − 4pM 2aP3E ð Þ − q ¼ þ 2 2 field. On each side of the surface, the magnetic ¼ e1 e2 1=m1 1=m2 4a P3 where P (x)=q(x)/2p is the magneto-electric ð þ Þð þ Þþ field induced by the surface current can be viewed 3 3 polarization (3), D is the electric displacement, P ð Þ as the field induced by an image magnetic mono- is the electric polarization, H is the magnetic We will first take e1 = e2 = m1 = m2 =1below pole on the opposite side. field, and M is the magnetization. It takes the and then recover the e1,2, m1,2 when discussing According to the above calculation, the image value of P3 =0invacuumorconventional the experimental proposals later. The solution magnetic monopole field indeed has the correct insulators and P3 = T1/2 in topological insulators, shows that, for an electric charge near the surface magnetic field dependence expected from a with the sign determined by the direction of the of a topological insulator, both an image mag- monopole, and it can be controlled completely surface magnetization. netic monopole and an image electric charge will through the position of the electric charge. As we Now consider the geometry as shown in Fig. 1. be induced, as compared with conventional elec- started with the Maxwell’s equation, which The lower-half space (z <0)isoccupiedbya tromagnetic media where only an electric image includes ∇⋅B 0, the magnetic flux integrated ¼ topological insulator with a dielectric constant e2 charge will be induced. It is notable that the over a closed surface must vanish. We can check and a magnetic permeability m2,whereasthe magnitudes of the image magnetic monopole and that this is the case by considering a closed upper-half space (z > 0) is occupied by a image electric charge satisfy the relation q1,2 = surface—for example, a sphere with radius a— conventional insulator with a dielectric constant T(aP3)g1,2.Thisisjusttherelationq =(q/2p)g for that encloses a topological insulator. The detailed e1 and a magnetic permeability m1.Apointelec- the electric and magnetic charges of a dyon inside calculation is presented in the supporting online tric charge q is located at (0, 0, d)withd>0.The the q vacuum (16), with q/2p = TP3 here. material (17). Inside the closed surface, there is Maxwell equations, along with the modified The physical origin of the image magnetic not only a image magnetic monopole charge, but constituent equations and the standard boundary monopole is understood by rewriting part of also a line of magnetic charge density whose Non-uniform : inducedconditions, constitute magnetic a complete boundary value the Maxwell equationsmonopole as integral exactly cancels the point image magnetic ✓ problem. To solve this problem, the method of monopole. However, when the separation be- ∇ B 2aP3d z ^n E 4 REPORTS images (15)canbeused.Weassumethat,inthe Â ¼ ð Þ Â ð Þ tween the electric charge and the surface (d)is lower-half space, the electric field is given by an with P3 = T1/2 the value for the topological in- much smaller than the spherical radius (a), the that sucheffective a general point definition charge q of/e1 aand topological an image charge sulator [in the above equation, ∇ is the derivative magnetic field is completely dominated by the q1 at (0, 0, d), whereas the magnetic field is given vector, ^n is the normal vector of the surface, and image magnetic monopole, and the contribution insulator reduces to the Z2 topological insulators Qi, Li, Zang & Zhang, describedby in an ( image4–6) formagnetic non-interacting monopole g band1 at (0, 0, d). d(z)istheDiracd function]. The right-hand side due to the line of magnetic charge density is Inducing a Magnetic Monopole with insulators;In this the upper-half finding is space, a three-dimensional the electric field is giv- of the above equation corresponds to a surface vanishingly small. Therefore, we propose here to en by q/e at (0, 0, d)andanimagechargeq at current density j sxy ^n E ,whichisinduced experimentally observe the magnetic monopole (3D) generalization1 of the quantum spin Hall 2 Science¼ ð Â Þ 323, 1148 (2009) insulator in two dimensions (7–10). For generic in the same sense that we can experimentally Topological Surface States band insulators, the parameter q has a micro- observe other fractionalization, or de-confinement, Fig. 2. Illustration of the experi- scopic expression of the momentum space phenomena in condensed matter physics. In any Xiao-Liang Qi,1 Rundong Li,1 Jiadong Zang,2 Shou-Cheng Zhang1 * mental setup to measure the image closed electronic system, the total charge must Chern-Simonsmonopole. form A (3, magnetic11). Recently, layer is experimental evidence of the topologically nontrivial be quantized to be an integer. However, one Existence of the magnetic monopole is compatible with the fundamental laws of nature; deposited on the surface of the surface states has been observed in Bi Sb alloy can separate fractionally charged elementary however, this elusive particle has yet to be detected experimentally. We show theoretically that an topological insulator, as indicated1−x x (12), which supports the theoretical prediction electric charge near a topological surface state induces an image magnetic monopole charge due by the layer with blue arrows. (The that Bi Sb is a Z topological insulator (4). to the topological magneto-electric effect. The magnetic field generated by the image magnetic 1−xsamex layer2 is drawn in Figs. 3 and With periodic temporal and spatial boundary monopole may be experimentally measured, and the inverse square law of the field dependence 4.) A scanning MFM tip carries a conditions,magnetic the partition flux f functionand a charge is periodicq.A in q can be determined quantitatively. We propose that this effect can be used to experimentally realize under thecharged 2p shift, impurity and the is confined system on is theinvariant a gas of quantum particles carrying fractional statistics, consisting of the bound states of the under thesurface time-reversal with charge symmetryQ and distance at q = 0 and electric charge and the image magnetic monopole charge. q = p.However,withopenboundaryconditions,D out of the surface. By scanning the partitionover function the voltage is noV and longer the distanceperiodic in q, he electromagnetic response of a conven- Furthermore, when a periodic boundary condi- and time-reversalr to the impurity, symmetry the is effect generally of the broken tional insulator is described by a dielectric tion is imposed in both the spatial and temporal (but onlyimage on the boundary), monopole evenmagnetic when fieldq =(2n + Tconstant e and a magnetic permeability m. directions, the integral of such a total derivative 1)p.Ourworkin(can be measured3)givesthefollowingphysical (see text). An electric field induces an electric polarization, term is always quantized to be an integer; i.e., interpretation: Time-reversal invariant topologi- S q cal insulators have a bulk energy gap but have Fig. 3. Illustration of the fractional statistics whereas a magnetic field induces a magnetic ℏ qn (where n is an integer). Therefore, the polarization. As both the electric field E(x)and partition¼ function and all physically measurable gapless excitations with an odd number of Dirac induced by image monopole effect. Each elec- the magnetic induction B(x)arewelldefinedinside quantities are invariant when the q parameter is cones on the surface. When the surface is coated tron forms a dyon with its image monopole. When two electrons are exchanged, an AB phase an insulator, the linear response of a convention- shifted by 2p times an integer (2). Under time- with a thin magnetic film, time-reversal sym- factor is obtained (which is determined by half of al insulator can be fully described by the effective reversal symmetry, eiqn is transformed into e–iqn metry is broken, and an energy gap also opens up 2 the image monopole flux) and leads to statistical 1 3 2 1 2 3 (here, i = –1). Therefore, all time-reversal in- at the surface. In this case, the low-energy theory transmutation. action S0 8p ∫d xdt eE − m B ,whered xdt is ¼ variant insulators fall into two general classes, is completely determined by the surface term in the volume element of space and time. However, described by either q =0orq = p (3). These two Eq. 1. As the surface term is a Chern-Simons in general, another possible term is allowed in the time-reversal invariant classes are disconnected, term, it describes the quantum Hall effect onwww.sciencemag.org SCIENCE VOL 323 27 FEBRUARY 2009 1185 effective action, which is quadratic in the elec- and they can only be connected continuously by the surface. From the general Chern-Simons- tromagnetic field, contains the same number of time-reversal breaking perturbations. This classi- Landau-Ginzburg theory of the quantum Hall derivatives of the electromagnetic potential, and fication of time-reversal invariant insulators in effect (13), we know that the coefficient q = is rotationally invariant; this term is given by terms of the two possible values of the q (2n +1)p gives a quantized Hall conductance of q a 3 e2 1 e2 Sq ∫d xdtE⋅B.Here,a (where ħ parameter is generally valid for insulators with sxy n . This quantized Hall effect on ¼ 2p 2p ¼ ℏc ¼ þ 2 h is PlanckÀÁ’sconstantÀÁ h divided by 2p and c is the arbitrary interactions (3). The effective action the surfaceÀÁ is the physical origin behind the speed of light) is the fine-structure constant, and q contains the complete description of the topological magneto-electric (TME) effect. can be viewed as a phenomenological parameter electromagnetic response of topological insula- Under an applied electric field, a quantized in the sense of the effective Landau-Ginzburg tors. Topological insulators have an energy gap in Hall current is induced on the surface, which in theory. This term describes the magneto-electric the bulk, but gapless surface states protected by turn generates a magnetic polarization and vice effect (1), where an electric field can induce a the time-reversal symmetry. We have shown (3) versa. magnetic polarization, and a magnetic field can induce an electric polarization. Unlike conventional terms in the Landau- Fig. 1. Illustration of the image charge Ginzburg effective actions, the integrand in Sq and monopole of a point-like electric is a total derivative term, when E(x) and B(x) charge. The lower-half space is occupied are expressed in terms of the electromagnetic by a topological insulator (TI) with di- vector potential (where ∂m denotes the partial electric constant e2 and magnetic perme- derivative; m, n, r, and t denote the spacetime ability m2.Theupper-halfspaceisoccupied coordinates; F mn is the electromagnetic field by a topologically trivial insulator (or vac- tensor; and Am is the electromagnetic potential) uum) with dielectric constant e1 and magnetic permeability m1. A point electric q a 3 mn rt charge q is located at (0, 0, d). When seen Sq ∫d xdtemnrtF F from the lower-half space, the image ¼ 2p 16p (1) q a 3 m n r t electric charge q1 and magnetic monopole ∫d xdt∂ emnrsA ∂ A ¼ 2p 4p ð Þ g1 are at (0, 0, d); when seen from the upper-half space, the image electric charge q2 and magnetic monopole g2 1Department of Physics, Stanford University, Stanford, CA are at (0, 0, −d). The red solid lines 94305–4045, USA. 2Department of Physics, Fudan Uni- represent the electric field lines, and blue versity, Shanghai, 200433, China. solid lines represent magnetic field lines. *To whom correspondence should be addressed. E-mail: (Inset) Top-down view showing the in-plane component of the electric field at the surface (red arrows) [email protected] and the circulating surface current (black circles).

1184 27 FEBRUARY 2009 VOL 323 SCIENCE www.sciencemag.org where do these “monopoles” come from? what about B =0 everywhere? r · where do these “monopoles” come from? what about B =0 everywhere? r · Authors use 2 image charges and 2 image monopoles where do these “monopoles” come from? what about B =0 everywhere? r · Authors use 2 image charges and 2 image monopoles For d=0 a direct solution of ﬁeld equations can be obtained

↵✓ 2 zˆ E(r0,z0 = 0) A(r,z)= d r0 ⇥ 4⇡2 (r r )2 + z2 Z 0 p

Nogueira & JvdB, arXiv:1808.08825 where do these “monopoles” come from? what about B =0 everywhere? r · Authors use 2 image charges and 2 image monopoles For d=0 a direct solution of ﬁeld equations can be obtained

↵✓ 2 zˆ E(r0,z0 = 0) A(r,z)= d r0 ⇥ 4⇡2 (r r )2 + z2 Z 0 ↵✓ 2 E(r0,z0 = 0) B(r,z)= A = z pd r0 r⇥ 4⇡2 [(r r )2 + z2]3/2 ⇢ Z 0 2 (r r0)+zzˆ zˆ d r0 E(r0,z0 = 0) [(r r )2 + z2]3/2 · Z 0 = Magnetic field satisfies B =0 Nogueira & JvdB, ) r · arXiv:1808.08825 where do these “monopoles” come from? what about B =0 everywhere? r · Authors use 2 image charges and 2 image monopoles For d=0 a direct solution of field equations can be obtained

Point vortex Nogueira & JvdB, arXiv:1808.08825 where do these “monopoles” come from? what about B =0 everywhere? r · Authors use 2 image charges and 2 image monopoles For d=0 a direct solution of ﬁeld equations can be obtained

Point vortex Monopole Nogueira & JvdB, arXiv:1808.08825 where do these “monopoles” come from? what about B =0 everywhere? r · Direct solution for ﬁnite d

1.0

0.8

0.6

0.4

0.2

Nogueira & JvdB, arXiv:1808.08825 Non-uniform ✓ : magnetic solenoid with flux B

B zˆ B ✓ @✓/@z

TI Non-uniform ✓ : magnetic solenoid with flux B e2 Gauss’ law E =4⇡⇢ ✓ B r · ⇡ r ·

B zˆ B ✓ @✓/@z

TI Non-uniform ✓ : magnetic solenoid with flux B e2 Gauss’ law E =4⇡⇢ ✓ B r · ⇡ r · total e2 d✓ Q = q + d2rB(r) dz charge ! 4⇡2 dz Z Z

B zˆ B ✓ @✓/@z

TI Non-uniform ✓ : magnetic solenoid with flux B e2 Gauss’ law E =4⇡⇢ ✓ B r · ⇡ r · total e2 d✓ e2✓ Q = q + d2rB(r) dz = q + charge ! 4⇡2 dz 4⇡2 B Z Z

B zˆ B ✓ @✓/@z

TI Non-uniform ✓ : magnetic solenoid with flux B e2 Gauss’ law E =4⇡⇢ ✓ B r · ⇡ r · total e2 d✓ e2✓ Q = q + d2rB(r) dz = q + charge ! 4⇡2 dz 4⇡2 B Z Z = q + q0

B zˆ B ✓ @✓/@z

TI Non-uniform ✓ : magnetic solenoid with flux B e2 Gauss’ law E =4⇡⇢ ✓ B r · ⇡ r · total e2 d✓ e2✓ Q = q + d2rB(r) dz = q + charge ! 4⇡2 dz 4⇡2 B Z Z = q + q0

B zˆ B ✓ @✓/@z

TI Non-uniform ✓ : magnetic solenoid with flux B e2 Gauss’ law E =4⇡⇢ ✓ B r · ⇡ r · total e2 d✓ e2✓ Q = q + d2rB(r) dz = q + charge ! 4⇡2 dz 4⇡2 B Z Z = q + q0

B zˆ B ✓ @✓/@z

TI + q0 Non-uniform ✓ : magnetic solenoid with flux B e2 Gauss’ law E =4⇡⇢ ✓ B r · ⇡ r · total e2 d✓ e2✓ Q = q + d2rB(r) dz = q + charge ! 4⇡2 dz 4⇡2 B Z Z MAGNETIC FLUX INDUCES CHARGE = q + q0

B zˆ B ✓ @✓/@z