A thin bimetallic strip of thickness d is straight at a temperature T0, with length L0. The two metals have coefficients of linear thermal expansion α1 and α2, with α2>α1. If the temperature is raised to a temperature T not significantly greater than T0, what is the angle θ through which the strip bends?
We should expect the curvature to be uniform, leaving the strip to form a circular arc of angle θ. If the radius of curvature for the inner half of the strip is r, then (with θ in radians), the length of that metal is L1=rθ.
We can ignore any expansion in the direction of the thickness, so the length of the other metal is
Taking the difference, we see
Now, the formula for thermal expansion is
(see this previous post),
so for the inner side,
and solving for L1,
Thus, the difference is
and so the angle θ is