The scattering of an arbitrary electromagnetic field by a conducting or non‐conducting obstacle is investigated. The differential equations and boundary conditions satisfied by the field are transformed into a pair of inhomogeneous linear integro‐differential equations forEandH. For an obstacle which is a thin shell of constant thicknessh, a formal procedure for obtaining a solution of these equations as power series inhis given. The lowest order term in this solution is the incident field. An explicit expression for the next term is found in the form of a surface integral. This integral is evaluated approximately by the method of stationary phase. The physical properties of the solution are examined in detail, and satisfactory agreement is found with many results previously obtained by other methods.