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 $Lambda = 2lambda^2/d$, 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.
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