Chapter 30, Problem 30.73CP

A wire carrying a current I is bent into the shape of an exponential spiral, r = e^θ, from θ = 0 to θ = 2π as suggested in Figure P29.47. To complete a loop, the ends of the spiral are connected by a straight wire along the x axis. (a) The angle β between a radial line and its tangent line at any point on a curve r = f(θ) is related to the function by

$\tan \beta =\frac{r}{dr/d\theta }$

Use this fact to show that β = π/4. (b) Find the magnetic field at the origin.

Figure P29.47