Calculating the Magnetic Field Due to a Current — Question 3

Question 3

Reshaping a fixed wire changes the central field despite an unchanged perimeter

A thin wire of total length P is first a square loop and then a circular loop. The same steady current I runs counterclockwise in each. Neglect the feed wires. For a straight segment use $B=\mu_0 I(\sin\theta_1+\sin\theta_2)/(4\pi a)$, with angles measured from its perpendicular bisector. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the central field of the square in terms of P.

2. Find the circular-loop field and the ratio of the two fields.

Question 3 - Solution

Reshaping a fixed wire changes the central field despite an unchanged perimeter

(a) Method and calculation

$$s=P/4,\quad a=s/2,\quad\theta_1=\theta_2=45^\circ.$$$$B_\square=4\frac{\mu_0 I\sqrt2}{4\pi(s/2)}=\boxed{\frac{8\sqrt2\mu_0 I}{\pi P}}.$$

(b) Interpretation and further calculation

$$R=P/(2\pi),\quad B_\circ=\frac{\mu_0 I}{2R}=\boxed{\frac{\pi\mu_0 I}{P}}.$$$$\frac{B_\square}{B_\circ}=\boxed{\frac{8\sqrt2}{\pi^2}}=1.14632.$$

Physical check

The circle encloses more area, but central field is not determined by area alone; the square sides pass closer to the center.