  In this paper I show how to calculate the effect of a nearby Pearl vortex or antivortex upon the critical current $I_c(B)$ when a perpendicular magnetic induction $B$ is applied to a planar Josephson junction in a long, thin superconducting strip of width $W$ much less than the Pearl length $\Lambda = 2\lambda^2/d$, where $\lambda$ is the London penetration depth and $d$ is the thickness ($d < \lambda$). The theoretical results provide a qualitative explanation of unusual features recently observed experimentally by Golod {\it et al.}\cite{Golod10} in a device with a similar geometry. 
Effect of nearby Pearl vortices upon the $I_c$ vs $B$ characteristics of   planar Josephson junctions in thin and narrow superconducting strips