A wire carrying a current I is bent into the shape of an exponential spiral r = e θ from θ = 0 to θ = 2 π , as in Fig.
To complete a loop, the ends of the spiral are connected by a straight wire along the x-axis. Find the magnitude and direction of 
at the origin.
Text Solution
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Sol. The angle between radial line & its tangent line at any point of the curve, r = f( θ) is related to the function in the following way
tan β = 
Thus in this case r = e θ , tan β = 1 & β = π /4
Therefore, the angle between 
& 
is
π – β = 3 π /4. Also
ds = 
= 
dr
using Biosavart law:
dB = 
= 
dB = 
B = 
Where r i = e 0 = 1
after integration
r f = e 2 π B = 
, out of page.
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