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the response of superconducting films to homogeneous applied magnetic fields is well understood , and analytic expressions for the distributions of the magnetic field and current density have been derived  @xcite based on bean s critical state model . @xcite when small current - carrying coils are placed near the surface to probe the local properties of superconducting films , the magnetic fields generated by the coils are inhomogeneous . analytic expressions describing the response of superconducting films to small coils have been derived for the linear response regime ,  @xcite but in order to measure the local distribution of the critical current density @xmath2 in superconducting films , it is necessary to investigate the nonlinear response . @xcite numerical computations of the nonlinear response of superconducting films to the inhomogeneous magnetic fields arising from small coils have been carried out in refs .  , but here we present analytic results for the nonlinear response to line currents above superconducting films .    the procedure proposed by claassen _ et al . _ @xcite for inductive measurements of the local @xmath2 distribution in films of thickness much greater than the london penetration depth @xmath7 can be described briefly as follows . a small coil carrying a sinusoidal drive current @xmath8 is placed just above a superconducting film , and the induced voltage @xmath9 in the coil is detected . the amplitude of the third - harmonic voltage @xmath10 is measured as a function of the drive current amplitude @xmath4 , and the threshold current @xmath11 is defined such that @xmath12 for @xmath13 and @xmath14 for @xmath15 . because @xmath6 , where @xmath3 the film thickness , @xmath2 can be evaluated from @xmath11 . @xcite since an electric - field criterion must be applied for a precise determination of @xmath2 , it is important to evaluate the electric field @xmath16 generated in the superconducting film .  @xcite    in the present paper we consider linear wires as simple models of coil wires , and we analytically investigate the response of a superconducting film to linear wires carrying transport currents . in sec . [ sec_single - wire ] we investigate the dc ( ac ) response of a superconducting film to a linear wire carrying a dc ( ac ) transport current : we determine the threshold current @xmath11 , and we present the voltage @xmath17 and the harmonic voltages induced in the linear wire , as well as the electric field @xmath16 induced in the superconducting film . in sec . [ sec_two - wires ] we consider a superconducting film and two linear wires carrying transport currents of opposite directions . we briefly summarize our results in sec . [ sec_conclusion ] . in this section we consider a superconducting film and a linear wire carrying a transport current , as shown in fig . [ fig_sc - wire ] .     and a linear wire at @xmath18 . the film is infinitely extended in the @xmath19 plane , and the infinite wire is parallel to the @xmath20 axis . ]    an infinitely long wire , parallel to the @xmath20 axis , is situated at @xmath18 where @xmath21 . the radius of the wire @xmath22 is assumed to be much smaller than @xmath23 . a superconducting film , infinitely extended in the @xmath19 plane , is situated at @xmath24 , where the film thickness @xmath3 is much smaller than @xmath23 but is larger than the london penetration depth @xmath7 . flux pinning in the film is characterized by the critical current density @xmath2 , which is assumed to be constant ( independent of magnetic field ) as in bean s critical state model  @xcite and to be spatially homogeneous in the film . we consider the limit @xmath25 , as this simplification allows us to obtain simple analytic expressions for the magnetic - field and current distributions . in the thin - film limit of @xmath25 , the sheet current @xmath26 plays crucial roles , and the upper limit of @xmath5 is the critical sheet - current density @xmath1 . the lower critical field @xmath27 is assumed to be much smaller than @xmath1 ( i.e. , @xmath28 ) , such that the flux penetration into superconducting films is not affected by @xmath27 , but is determined by @xmath1 . @xcite    we introduce the complex field @xmath29 ,  @xcite which is an analytic function of @xmath30 for @xmath31 and @xmath32 . the biot - savart law for the complex field is given by @xmath33 where @xmath34 is the complex field arising from the line current alone . the @xmath34 is given by @xmath35 where @xmath36 is the transport current flowing in the linear wire . at the upper ( @xmath37 ) and lower ( @xmath38 ) surfaces of the superconducting film , where @xmath39 is a positive infinitesimal , the perpendicular and parallel magnetic fields @xmath40 and @xmath41 are obtained from eq .   as @xmath42 where p denotes the cauchy principal value integral . the complex potential is defined by @xmath43 , and the contour lines of the real part of @xmath44 correspond to magnetic - field lines . the magnetic flux per unit length @xmath45 around the linear wire is @xmath46 . \label{phi - wire_def}\end{aligned}\ ] ] we have introduced a cutoff length @xmath47 , where @xmath48 is of the order of the radius of the wire , to remove the logarithmic divergence as @xmath49 . the magnetic flux per unit length @xmath50 up through the film ( @xmath51 ) in the region @xmath52 is @xmath53 . \label{phi - film_def}\end{aligned}\ ] ]      in this subsection we consider the magnetic - field distribution when the linear wire carries a dc current @xmath54 that is held constant after monotonically increasing from @xmath55 .      for @xmath56 , the magnetic field is completely shielded below the film , @xmath57 . the field distribution can be obtained by the mirror - image technique , and the resulting complex field is @xmath58 the complex potential @xmath43 for @xmath59 is given by @xmath60 the perpendicular magnetic field and sheet - current density are thus given by @xmath61 and @xmath62 respectively . the net current induced in the superconducting film is @xmath63 , as expected . note that the sheet - current density can not exceed @xmath1 ; that is , @xmath64 . because the maximum of @xmath5 given by eq . is @xmath65 , eq . is valid for @xmath66 , where the threshold current is given by @xmath67 figure  [ fig_field - lines](a ) shows the magnetic - field lines [ i.e. , the contour lines of @xmath68 calculated from eq .  , and the dashed line in fig . [ fig_field - lines](d ) shows @xmath69 given by eq ..        the magnetic flux per unit length around the linear wire , calculated by substituting eq .   into eq .  , is @xmath70 where the inductance per unit length @xmath71 corresponds to the difference between the self inductance of the linear wire and the mutual inductance of the wire and its image . because the perpendicular magnetic field in the film is zero , the magnetic flux up through the film defined by eq . is also zero :  @xcite @xmath72    equations  , , , , and are valid for @xmath56 .      for @xmath15 , on the other hand , the maximum @xmath5 reaches @xmath1 and the magnetic field penetrates below the superconducting film . the field distributions for @xmath15 , therefore , must satisfy @xmath73 where the flux fronts [ i.e. , the boundaries between the region of @xmath74 and that of @xmath61 ] are at @xmath76 . the complex field and the parameter @xmath77 are determined such that they are consistent with eqs .   and , as derived in appendix [ app_deriv - ch ] . the parameter @xmath77 is a function of @xmath4 , @xmath78 and the complex field is @xmath79 where the upper sign holds for im@xmath80 and the lower sign for im@xmath81 . the corresponding complex potential is @xmath82\right\ } . \nonumber\\ \label{cg_nonlinear}\end{aligned}\ ] ] figures  [ fig_field - lines](b ) and ( c ) show the magnetic - field lines , i.e. , the contour lines of @xmath83 calculated from eq .  .    the perpendicular magnetic field @xmath40 in the superconducting film , obtained from eq , is @xmath84          0 & \mbox{for } |x|>a ,       \end{cases } \label{hy_nonlinear}\ ] ] and the sheet - current density @xmath85 $ ] is @xmath86          \displaystyle -j_cd \left(1           -\frac{|x|\sqrt{x^2-a^2}}{x^2+y_0 ^ 2 } \right )           & \mbox{for } |x|>a . \end{cases } \nonumber\\ \label{kz_nonlinear}\end{aligned}\ ] ] the net current induced in the superconducting film is again @xmath63 , as expected . figures  [ fig_field - lines](e ) and ( f ) show @xmath87 and @xmath69 given by eqs .   and , respectively .    the magnetic flux per unit length around the linear wire , calculated by substituting eq . into eq .  , is @xmath88 . \label{phi - wire_nonlinear}\ ] ] combining eqs .  , , and yields @xmath89 , where @xmath90          \displaystyle          l_0i_0 + \frac{\mu_0}{2\pi } \left [ i_{c0}-i_0+i_0           \ln\left(\frac{i_0}{i_{c0}}\right)\right ]           & \mbox{for } i_0>i_{c0 } . \end{cases } \label{phi_dc - def}\end{aligned}\ ] ] the magnetic flux up through the film , calculated by substituting eq .   into eq . , is @xmath91   \label{phi - film_nonlinear}\ ] ] for @xmath92 , and @xmath93 for @xmath94 . combining eqs .  , , and yields @xmath95 , where @xmath96          \displaystyle          \frac{\mu_0}{2\pi } \left [           \sqrt{i_0 ^ 2 -i_{c0}^2\left(1+\frac{x^2}{y_0 ^ 2}\right ) }           + i_0\mbox{arccosh } \left(\frac{i_0}{i_{c0 } }           \frac{y_0}{\sqrt{x^2+y_0 ^ 2}}\right ) \right ]           & \mbox{for } i_0>i_{c0 } \mbox { and } |x|<a . \end{cases } \label{phi - film_dc - def}\end{aligned}\ ] ]      in this subsection we consider the time - dependent field distributions when the linear wire carries a sinusoidal ac drive current @xmath97 . in inductive measurements of the local @xmath2 in superconducting films , harmonic voltages induced in coils are detected . @xcite for precise @xmath2 measurements , it also is important to determine the electric field induced in the film . @xcite    we wish to calculate the magnetic field around the linear wire @xmath98 , defined by eq .  , and the voltage per unit length induced in the wire , @xmath99 where @xmath100 is the resistance per unit length of the wire . we also wish to determine the magnetic flux per unit length up through the film @xmath101 , defined by eq .  , and the electric field in the film,@xcite @xmath102      for @xmath56 the magnetic field is completely shielded below the superconducting film , @xmath57 , as in sec . [ sec_dc - linear ] . the complex field , the sheet - current density in the film , and the magnetic flux around the linear wire are given by eqs .  , , and , respectively , except that now @xmath4 in those equations is replaced by @xmath8 . the magnetic flux per unit length around the linear wire is given by @xmath103 . the voltage induced in the wire defined by eq . is thus given by @xmath104 . for @xmath105 , the harmonic voltages , the magnetic flux per unit length penetrating the film @xmath106 , and the electric field in the film @xmath16 are all zero . @xcite      for @xmath15 , on the other hand , the magnetic field penetrates through the superconducting film , as discussed in sec . [ sec_dc - nonlinear ] .    for ac drive current @xmath8 , the magnetic flux per unit length around the linear wire @xmath98 is  @xcite @xmath107 for @xmath108 , and @xmath109 for @xmath110 , where @xmath111 is defined by eq .  . the voltage per unit length of the wire , calculated from eqs .   and , is @xmath112 where @xmath113 is the differential inductance given by @xmath114          \displaystyle           l_0 + \frac{\mu_0}{2\pi } \ln\left(\frac{i_0}{i_{c0}}\right )           & \mbox{for } i_0>i_{c0 } . \end{cases } \label{l_dc - def}\ ] ]    in response to the ac drive current , the magnetic flux per unit length up through the film @xmath101 is  @xcite @xmath115 for @xmath108 , and @xmath116 for @xmath110 , where @xmath117 is defined by eq .  . the electric field induced in the film , calculated from eqs .   and , is @xmath118 where the function @xmath119 is given by @xmath120          \displaystyle           \frac{\mu_0}{2\pi } \mbox{arccosh } \left(\frac{i_0}{i_{c0 } }           \frac{y_0}{\sqrt{x^2+y_0 ^ 2}}\right ) \\[2ex ]          \hfill\mbox{for } i_0>i_{c0 } \mbox { and } |x|<a . \end{cases } \label{l - film_dc - def}\ ] ] in order to measure @xmath2 in superconducting films by the inductive method detecting the harmonic voltages ,  @xcite it is important to estimate the induced electric field in superconducting films . @xcite the maximum electric field is induced just below the linear wire at @xmath121 , and is given approximately by @xmath122 for @xmath123 . the voltage per unit length induced in the linear wire can be expressed as the fourier series @xmath124 where the amplitude @xmath125 and phase difference @xmath126 of the @xmath127th harmonics are calculated by @xmath128 because of the periodicity of @xmath129 , the even harmonics are zero ; i.e. , @xmath130 for @xmath131 . the fundamental voltage @xmath132 is simply determined by the resistance per unit length of the linear wire , the self - inductance per unit length of the wire , and the mutual inductance per unit length between the wire and its image . we are interested in the odd harmonics of @xmath133 for @xmath134 ; they are obtained by substituting eqs .   and into eq .  , which yields @xmath135 where @xmath136 . for@xmath137 we have @xmath138    figure  [ fig_v3-q3-i0 ] shows the @xmath4 dependence of the third - harmonic voltage @xmath139 calculated from eq .   with @xmath140 . although the present model of a linear wire is oversimplified , the behavior of the third harmonic voltage shown in this figure qualitatively agrees with the experimental data measured by a coil with a @xmath141 film . @xcite     and the phase @xmath142 of the third - harmonic voltage as functions of the drive current amplitude @xmath4 . ] two linear wires are better than a single wire to model current - carrying coils . in this section we consider a superconducting film and two linear wires carrying transport currents of opposite directions , as shown in fig . [ fig_sc-2wires ] .     and two linear wires at @xmath143 carrying transport currents @xmath144 . ] infinitely long wires , parallel to the @xmath20 axis , are situated at @xmath143 where @xmath145 and @xmath21 . a superconducting film , infinitely extended in the @xmath19 plane , is situated at @xmath24 , where @xmath146 . one wire carrying a dc current @xmath147 is situated at @xmath148 and another wire carrying a current of opposite direction @xmath149 is at @xmath150 , where @xmath145 and @xmath21 . the complex field due to the two wires is given by @xmath151 where @xmath152 and @xmath153 . in this subsection we consider the magnetic - field distribution when the linear wire carries a dc current @xmath54 that is held constant after monotonically increasing from @xmath55 .      for @xmath56 , the magnetic field is completely shielded below the film , @xmath57 . the field distribution can be obtained by the mirror - image technique , and the resulting complex field is @xmath154 where @xmath155 the complex potential for @xmath156 is given by @xmath157    the perpendicular magnetic field is @xmath61 , and the sheet - current density is thus given by @xmath158 where the function @xmath159 is determined by the configuration of the wires , @xmath160 .   \label{kz / i0_2wires}\ ] ] the sheet - current density @xmath161 is maximum at @xmath162 , where @xmath163 the sheet - current density can not exceed @xmath1 ; that is , @xmath164 , and eq . is valid for @xmath66 , where the threshold current is given by @xmath165    figure  [ fig_field - lines-2wires](a ) shows the magnetic - field lines [ i.e. , the contour lines of @xmath68 calculated from eq .  , and the dashed line in fig .  [ fig_field - lines-2wires](d ) shows @xmath69 given by eq .  .          for @xmath15 , on the other hand , the maximum @xmath5 reaches @xmath1 and the magnetic field penetrates below the superconducting film . the field distributions for @xmath15 , therefore , must satisfy @xmath166 where the flux fronts are at @xmath76 and @xmath167 . the complex field and the parameters @xmath77 and @xmath168 ( where @xmath169 ) are determined such that they are consistent with eqs .   and , as derived in appendix [ app_two - wires - xz ] . the parameters for flux fronts , @xmath77 and @xmath168 , are determined as functions of @xmath4 for @xmath15 , by solving the following two equations , @xmath170 ,   \label{ab - i0_2}\end{aligned}\ ] ] where @xmath171 and @xmath172 are the complete elliptic integrals of the first and second kinds , and @xmath173 . these parameters obey @xmath174 when @xmath175 , and @xmath176 when @xmath15 , where @xmath177 is given by eq .  . as shown in fig . [ fig_2wires - ab - i0 ] , @xmath77 increases and @xmath168 decreases with increasing @xmath4 . ( upper lines ) and @xmath168 ( lower lines ) , as functions of @xmath4 , determined by eqs .   and . the solid lines show @xmath77 and @xmath168 for @xmath178 ( @xmath179 ) , the dashed lines for @xmath180 ( @xmath181 ) , and the chained lines for @xmath182 ( @xmath183 ) . ] the complex field is derived in appendix [ app_two - wires - xz ] , and is given by @xmath184 where @xmath185 ,   \nonumber\\ \label{hperp(z)}\\      { \cal h}_c(\zeta ) & = & \frac{j_cd}{\pi } \zeta\phi(\zeta ) \int_b^a           \frac{du}{(u^2-\zeta^2)\phi(u ) } ,   \label{hc(z)}\\      \phi(\zeta ) & = & \sqrt{(a^2-\zeta^2)(\zeta^2-b^2 ) } .   \label{phi(z)}\end{aligned}\ ] ] integrating eq .   [ or ] , we obtain the complex potential @xmath43 , as @xmath186   \nonumber\\      & & { } + \frac{j_cd}{\pi}\int_b^a du g_0(\zeta , u ) ,   \label{g(z)}\end{aligned}\ ] ] where @xmath187    figure  [ fig_field - lines-2wires](b ) and ( c ) show the magnetic - field lines , i.e. , the contour lines of @xmath83 calculated from eq .  . figure  [ fig_field - lines-2wires](e ) and ( f ) show @xmath40 and @xmath188 $ ] calculated from eq . [ or from eqs .  , , , and ] . the response of a superconducting film to ac drive currents @xmath189 flowing in the two linear wires at @xmath143 is calculated in a manner similar to that described in sec . [ sec_ac ] . the magnetic flux per unit length @xmath45 linked in the two wires is calculated from @xmath190 ,   \label{phi - wire_def-2}\end{aligned}\ ] ] and the magnetic flux per unit length @xmath50 up through the film ( @xmath51 ) in the region @xmath52 is calculated from eq .  , where @xmath44 is given by eqs .   and . the time - dependent magnetic flux is given by eqs .   and . the voltage @xmath17 induced in the wires per unit length and the electric field induced in the film are given by eqs .   and , respectively . here we discuss the more realistic situation of experiments detecting the response of a superconducting film to a circular current - carrying coil . @xcite we consider a single - turn coil parallel to a superconducting film . when the radius of the coil is @xmath191 and the distance between the coil and the film is @xmath23 , the configuration of the coil and the film in the @xmath192 plane is similar to that in fig . [ fig_sc-2wires ] . we expect that the magnetic - field lines , the magnetic - field component perpendicular to the film , and the circular sheet - current distribution in the film then will be similar to the corresponding quantities shown in fig .  [ fig_field - lines-2wires ] .    in actual superconducting films the critical current density @xmath2 and the film thickness @xmath3 can be inhomogeneous , although in the present paper we assumed that @xmath193 is spatially homogeneous . the magnetic field produced by the coil of radius @xmath191 is largest directly below the coil ( i.e. , the circular region of radius @xmath194 in the film ) when @xmath195 . if @xmath193 in the circular region below the coil is nearly homogeneous , the response of the film will be similar to that presented in sec . [ sec_two - wires ] , and the signal in the coil will yield the local value of @xmath193 . in this sense the resolution of the measurements using the coil is on the order of @xmath191 . however , if @xmath193 is very inhomogeneous within the circular region below the coil , the response of the film will be quite different from that given in sec . [ sec_two - wires ] . we investigated the response of an infinite superconducting film at @xmath51 to linear wires above the film carrying transport currents @xmath36 . we derived analytic expressions for the complex field , and these are given in eqs . , , , and . the behavior of the film can be summarized as follows . \(i ) response to a dc current , @xmath196 : for a small dc current in the range @xmath197 , the sheet - current density @xmath0 in the film is less than the critical value @xmath1 , and no magnetic field penetrates into the region @xmath198 below the film . however , for a large dc current @xmath15 , the sheet - current density reaches @xmath199 and the magnetic field penetrates into the region below the film . \(ii ) response to an ac current , @xmath200 : for @xmath13 the voltage @xmath17 induced in the linear wire has only the first harmonic at the fundamental frequency @xmath201 . however , for @xmath15 , odd - harmonic voltages with frequency @xmath202 , @xmath127 = 3 , 5 , 7 , ... , are also induced in the linear wire . the complex field for the case of a single current - carrying linear wire above the film is derived in appendix a. the complex field for the case of a pair of current - carrying wires in a plane perpendicular ( parallel ) to the film is given in appendix b ( appendix c ) . we thank h.  yamasaki for stimulating discussions . this manuscript has been authored in part by iowa state university of science and technology under contract no . w-7405-eng-82 with the u.s . department of energy . in the @xmath203 plane for the integrals in eqs .   and consists of a line just above the real axis , @xmath204 from @xmath205 to @xmath206 where @xmath207 , an infinite circle , @xmath208 from @xmath209 to @xmath210 , a line just below the real axis , @xmath211 from @xmath206 to @xmath205 , and an infinitesimal line parallel to the imaginary axis , @xmath212 from @xmath213 to @xmath214 , taking the limit of @xmath215 , @xmath216 , and @xmath217 . ]    in this appendix we derive eqs .   and . consider the function @xmath218 defined by @xmath219           \sqrt{a^2-\zeta^2 } ,   \label{f(z)}\ ] ] where @xmath220 is @xmath221 we calculate the integral @xmath222 along the closed contour @xmath223 shown in fig . [ fig_contour ] , as @xmath224 where the contour integral along the infinite circle vanishes because @xmath225 for @xmath226 . because the integrand in the left - hand side of eq . has poles at @xmath227 and @xmath228 , the contour integral of eq . is also calculated by using the residue theorem , as @xmath229 , and thus yielding @xmath230 where @xmath231 is given by @xmath232    for @xmath233 , the @xmath234 and @xmath220 are given by @xmath235 ,       \hspace{3ex } \label{h(x+i0)}\\      \eta_{\parallel}(x ) & = & i\,\mbox{im}\,{\cal h}_0(x ) ,   \label{hpara(x)}\end{aligned}\ ] ] respectively , and thus yielding @xmath236 by using eqs .  , , and @xmath237          \sqrt{a^2-x^2 } & \mbox{for } |x|<a ,       \end{cases}\ ] ] we have @xmath238          -i \sqrt{a^2-x^2 } k_z(x )           & \mbox{for } |x|<a ,       \end{cases } \nonumber\\      & = &       \begin{cases }          0 & \mbox{for } |x|>a , \\[1ex ]          i \sqrt{a^2-x^2 } j_cd           & \mbox{for } |x|<a ,       \end{cases } \nonumber\\ \label{f(x+i0)-f(x - i0)}\end{aligned}\ ] ] where we used eqs .   and . substitution of eq .   into eq .   yields @xmath239 where the upper ( lower ) signs hold for @xmath59 ( @xmath240 ) .    from eqs .  , , and we obtain @xmath241 note that the first term of the right - hand side of eq .  , which is proportional to @xmath4 , corresponds to the complex field in the ideal meissner state for two semi - infinite strips situated at @xmath242 with a line current at @xmath243 . by using @xmath244 , we confirm that eq .   for @xmath245 is identical to eq .  . equation   can be rewritten as @xmath246 however , the last term of the right - hand side of eq .   must vanish , because @xmath234 is finite at @xmath247 . we thus obtain @xmath248 , such that the parameter @xmath77 describing the position of the flux front is given by eq .  . the resulting expression for the complex field is given by eq .  . see also appendix [ app_two - wires ] for the complex field for a superconducting film with two wires in the @xmath249 plane . we present here the complex field for a superconducting film with two parallel linear wires above it . an infinite superconducting film is in the plane @xmath51 , as shown in fig . [ fig_sc - wire ] . one wire carrying a dc current @xmath250 is situated at @xmath251 and another wire carrying current @xmath252 is at @xmath253 , where @xmath254 . the derivation of the complex field for the two wires is similar to that given in appendix  [ app_deriv - ch ] for a single wire , and we exhibit here only the resulting expressions .    for @xmath13 , the sheet - current density and the perpendicular magnetic field satisfy @xmath255 and @xmath61 , respectively . the threshold current is given by @xmath256 and the complex field for @xmath13 is @xmath257 for @xmath15 , @xmath69 and @xmath87 fulfill eqs .   and , and the relationship between @xmath77 and @xmath4 is given by @xmath258 the complex field and complex potential for @xmath15 are respectively given by @xmath259 ,   \label{ch_two - wires}\\      { \cal g}(\zeta ) & = & \frac{j_cd}{2 }           \left(\pm i\zeta + \sqrt{a^2-\zeta^2 } \right )   \nonumber\\          & & { } + \frac{i_0}{2\pi }           \left [ i\arctan\left(\frac{\zeta}{y_1}\right )           -\mbox{arctanh}\left(\sqrt{%          \frac{a^2-\zeta^2}{a^2+y_1 ^ 2}}\right )           \right . \nonumber\\ & & \left . { }          -i\arctan\left(\frac{\zeta}{y_2}\right )           + \mbox{arctanh}\left(\sqrt{%          \frac{a^2-\zeta^2}{a^2+y_2 ^ 2}}\right ) \right ] . \nonumber\\ \label{cg_two - wires}\end{aligned}\ ] ] in this appendix we derive eqs .  . consider the function @xmath260 defined by @xmath261           \phi(\zeta ) ,   \label{f(z)}\ ] ] where @xmath262 and @xmath263 are defined by eqs .   and , respectively . we calculate the integral @xmath264 along the closed contour @xmath223 shown in fig . [ fig_contour ] , as in eq .  , @xmath265 where the contour integral along the infinite circle vanishes because @xmath266 for @xmath226 . because the integrand on the left - hand side of eq .   has poles at @xmath227 , @xmath267 , and @xmath268 , the contour integral of eq .   can be calculated by using the residue theorem , as @xmath269 , and thus yielding @xmath270 the @xmath271 is defined by @xmath272   \nonumber\\      & = & \frac{i_0}{2\pi }           \left [ \frac{\zeta\phi(\zeta_1)}{\zeta^2-\zeta_1 ^ 2 }           -\frac{\zeta\phi(\zeta_2)}{\zeta^2-\zeta_2 ^ 2 } \right ] ,   \label{f0(z)}\end{aligned}\ ] ] where we used @xmath273 and @xmath274 .    for @xmath233 , eq . is reduced to @xmath275          \mbox{sgn}(x ) \widetilde{\phi}(x ) & \mbox{for $ b<|x|<a$ } ,       \end{cases } \label{phi(x+i0)}\ ] ] where @xmath276          \sqrt{(a^2-x^2)(x^2-b^2 ) } & \mbox{for $ b<|x|<a$ } , \\[1ex ]          -\sqrt{(x^2-a^2)(x^2-b^2 ) } & \mbox{for $ |x|>a$ } . \end{cases } \nonumber\end{aligned}\ ] ] substituting eqs .   and into eq .   with @xmath233 , we have @xmath277          -i\,\mbox{sgn}(x ) \widetilde{\phi}(x ) k_z(x )           & \mbox{for $ b<|x|<a$ } ,           \end{cases } \nonumber\\      & = & \begin{cases }          0           & \mbox{for $ |x| < b$ or $ |x|>a$ } , \\[1ex ]          i\,j_cd\ , \widetilde{\phi}(x )           & \mbox{for $ b<|x|<a$ } ,           \end{cases } \label{f(x+i0)-f(x - i0 ) } \nonumber\\\end{aligned}\ ] ] where we used eqs .   and . substitution of eq .   into eq . yields @xmath278 such that @xmath279 by using @xmath280 we rewrite eq .   as @xmath281 \right . \nonumber\\      & & \left.\quad{}+\frac{j_cd}{\pi } \int_b^a           \frac{du}{(u^2-\zeta^2)\phi(u ) } \right\ }   \nonumber\\      & & { } + \frac{\zeta}{\phi(\zeta)}{\cal r}(\zeta ) ,   \label{h(z)-calc}\end{aligned}\ ] ] where @xmath282   \nonumber\\      & & { } -\frac{j_cd}{\pi}\int_b^a du           \frac{\zeta^2+u^2-a^2-b^2}{\widetilde\phi(u ) }   \nonumber\\      & = & \frac{(\zeta^2-b^2){\cal r}(\pm a )           + ( a^2-\zeta^2){\cal r}(\pm b)}{a^2-b^2 } . \label{r(z)}\end{aligned}\ ] ] for @xmath247 and @xmath283 , we have @xmath284 in order to remove the divergences in eq .   at @xmath247 and @xmath283 , we require @xmath285 , thus yielding @xmath286 equations   and reduce to eqs .   and , respectively . we thus obtain @xmath287 for any @xmath288 , and eq . is reduced to eqs .  , , and . the sheet current @xmath289 $ ] and the perpendicular magnetic field @xmath40 are obtained by substituting @xmath233 in eq .   with eq . , @xmath290\right .   \nonumber\\      & & \left.{}-\frac{j_cd}{\pi}\int_b^a           \frac{du}{[(x\pm i\epsilon)^2-u^2]\phi(u ) } \right\ }   \nonumber\\      & = & \phi(x\pm i\epsilon ) \left\ { \frac{i_0}{\pi }           \mbox{re}\,\left [ \frac{x}{(x^2-\zeta_1 ^ 2)\phi(\zeta_1 ) }           \right]\right . \nonumber\\      & & { } -\frac{j_cd}{\pi}\int_b^a \frac{du}{\phi(u ) }           \left[\frac{x}{x^2-u^2 } \right . \nonumber\\      & & \left.\left.\quad{}\mp\frac{i\pi}{2 }           \bigl(\delta(x - u)+\delta(x+u)\bigr)\right ] \right\ } . \label{hykz - cal}\end{aligned}\ ] ] substitution of eq .   into eq . yields the sheet current @xmath69 and the perpendicular magnetic field @xmath87 . the @xmath69 for @xmath291 is given by eq . , and that for @xmath292 or @xmath293 is given by @xmath294 \right . \nonumber\\      & & \left . { } -j_cd \int_{b}^{a }           \frac{du}{(x^2-u^2)\phi(u ) } \right\ } .   \label{kz_nonlinear_shld_2}\end{aligned}\ ] ] the @xmath87 for @xmath292 or @xmath293 is given by eq . , and that for @xmath291 is given by @xmath295 \right . \nonumber\\      & & \left . { } -j_cd\ , \mbox{p } \int_{b}^{a }           \frac{du}{(x^2-u^2)\phi(u ) } \right\ } ,   \label{hy_nonlinear_pntr_2}\end{aligned}\ ] ] where p denotes the cauchy principal value integral . 99 g. w. swan , j. math . * 9 * , 1308 ( 1968 ) . m. r. halse , j. phys . d * 3 * , 717 ( 1970 ) . p. n. mikheenko and yu . e. kuzovlev , physica c * 204 * , 229 ( 1993 ) . j. zhu , j. mester , j. lockhart and j. turneaure , physica c * 212 * , 216 ( 1993 ) . e. h. brandt , m. v. indenbom and a. forkl , europhys lett . * 22 * , 735 ( 1993 ) ; 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t. nadami , e. s. otabe , m. kiuchi , and t. matsushita , ibid . * 412 - 414 * , 1011 ( 2004 ) ; t. nadami , e. s. otabe , m. kiuchi , t. matsushita , y. mawatari , h. yamasaki , and y. nakagawa , ibid . * 426 - 431 * , 688 ( 2005 ) . h. yamada , a. bitoh , y. mitsuno , i. imai , k. nomura , k. kanayama , s. nakagawa , y. mawatari , and h. yamasaki , physica c * 433 * , 59 ( 2005 ) . m. aurino , e. di gennaro , f. di iorio , a. gauzzi , g. lamura , and a. andreone , j. appl . phys . * 98 * , 123901 ( 2005 ) . the flux - penetration process for finite @xmath296 is more complicated than that shown in the present paper for @xmath28 . the case for @xmath296 will be published elsewhere , j. r. clem and y. mawatari , unpublished . j. r. clem , r. p. huebener , and d. e. gallus , j. low temp . * 12 * , 449 ( 1973 ) . e. zeldov , a. i. larkin , v. b. geshkenbein , m. konczykowski , d. majer , b. khaykovich , v. m. vinokur , and h. shtrikman , , 1428 ( 1994 ) . y. mawatari and j. r. clem , phys . * 86 * , 2870 ( 2001 ) ; a. a. babaei brojeny , y. mawatari , m. benkraouda , and j. r. clem , supercond . * 15 * , 1454 ( 2002 ) ; y. mawatari and j. r. clem , , 024505 ( 2003 ) . note that the simple result of @xmath297 for @xmath13 is ideally accurate only for the thin - film limit , @xmath25 . small but finite magnitudes of @xmath298 and @xmath299 arise even for @xmath13 in a superconducting film of finite thickness , because the magnetic flux penetrates in the vicinity of the upper surface of the film . we confine our attention here to low frequencies , well below those normally used to investigate the surface resistance of superconducting films , where energy dissipation arises due to excitation of quasiparticles or to small - amplitude vortex motion , as discussed , for example , in m. w. coffey and j. r. clem , , 386 ( 1991 ) .
we theoretically investigate the response of a superconducting film to line currents flowing in linear wires placed above the film , and we present analytic expressions for the magnetic - field and current distributions based on the critical state model . 
 the behavior of the superconducting film is characterized by the sheet - current density @xmath0 , whose magnitude can not exceed the critical value @xmath1 , where @xmath2 is the critical current density and @xmath3 is the thickness of the film . 
 when the transport current @xmath4 flowing in the wire is small enough , @xmath5 is smaller than @xmath1 and the magnetic field is shielded below the film . 
 when @xmath4 exceeds a threshold value @xmath6 , on the other hand , @xmath5 reaches @xmath1 and the magnetic field penetrates below the film . 
 we also calculate the ac response of the film when an ac transport current flows in the linear wires .