Wednesday Homework Problem 8.13 The wire shown at the right carries a current of

Question

Wednesday Homework Problem 8.13 The wire shown at the right carries a current of A. The seg- ment of the wire is L 30cm long. Consider the magnetic field at the point P a distance d 6em from the end of the wire as drawn. Naturally all fields below must be reported as vectors. Carefully and clearly show all intermediate steps to your cal culations. (a)Compute the magnetic field at P using the infinite straight wire approximation. (b)Compute the magnetic field at point P ap- proximating the segment with one segment using the finite current element approxima- tion. starting from the Biot-Savart Law. in parts (a) and (b) to be very accurate? (c)Calculate the exact magnetic field at P (d)Do you expect either of the approximations Why?

Explanation / Answer

from the given data

current, I = 0.75 A

L = 30 cm

d = 6 cm

a. using infinite wire approximation

magnetic field at P will be

B = -2kI/d z' ( where z' is unit vector along z axis)

B = -2*10^(-7)*0.75/0.06 = -0.0000025 z' Tesla = -2.5*10^-6 z' T

b. considering the wire to be one segment

magnetic field at point P from biot savart law approximation is

B = k*I*L z'/(d^2 + L^2/4)

B = 10^-7*0.75*0.3 z'/(0.06^2 + 0.3^2/4)

B = -8.620689*10^-7 z' tesla

c. exact magnetic field at point P using biot savarts law is

B = kI(L/sqrt(L^2 + d^2))/d z' = kIL/sqrt(L^2 + d^2)d z'

B = 10^-7*0.75*0.3/sqrt(0.3^2 + 0.06^2)*0.06 z' = -4.00641*10^-6 z' T

d. as we can see, the result of part a is closer to the exact result as compared to the result of the part B, because L > d and hence when the point P is very close to the wire, we can consider the wire to be an infinite wire to get the final results

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