Question

1. Square current loop (10+10+10= 30 points) Consider a square wire loop of side length $b = 0.2$ m that lies in the $xy$ plane and is centered at the origin. It carries a current $I = 1$ A that flows counterclockwise if viewed from the positive $z$ axis. (a) Use symmetry considerations to determine the direction of the magnetic field on the $z$ axis. Note that it is useful to include, in addition to spatial operations like rotations and mirrors, also the time-reversal operation $T$, which reverses the current and the magnetic field. Note that the symmetry operations form a group, so one can also compose them like $m'_{xy} = Tm_{xy}$. (b) Find the magnitude of the exact magnetic field $|B|$ at position $(x, y, z) = (0, 0, 0.4)$ m. Use units of $10^{-6} T = mu T$ and state the result to three significant figures. (c) Find the magnitude of the magnetic field within the dipole approximation $|B_{dip}|$ at the same position $(x, y, z) = (0, 0, 0.4)$ m. Use units of $10^{-6} T = mu T$ and state the result to three significant figures.

          1. Square current loop
(10+10+10= 30 points)
Consider a square wire loop of side length $b = 0.2$ m that lies in the $xy$ plane and is
centered at the origin. It carries a current $I = 1$ A that flows counterclockwise if viewed
from the positive $z$ axis.
(a) Use symmetry considerations to determine the direction of the magnetic field on
the $z$ axis. Note that it is useful to include, in addition to spatial operations like
rotations and mirrors, also the time-reversal operation $T$, which reverses the current
and the magnetic field. Note that the symmetry operations form a group, so one
can also compose them like $m'_{xy} = Tm_{xy}$.
(b) Find the magnitude of the exact magnetic field $|B|$ at position $(x, y, z) = (0, 0, 0.4)$ m.
Use units of $10^{-6} T = mu T$ and state the result to three significant figures.
(c) Find the magnitude of the magnetic field within the dipole approximation $|B_{dip}|$
at the same position $(x, y, z) = (0, 0, 0.4)$ m. Use units of $10^{-6} T = mu T$ and state
the result to three significant figures.

1. Square current loop
(10+10+10= 30 points)
Consider a square wire loop of side length b = 0.2 m that lies in the xy plane and is
centered at the origin. It carries a current I = 1 A that flows counterclockwise if viewed
from the positive z axis.
(a) Use symmetry considerations to determine the direction of the magnetic field on
the z axis. Note that it is useful to include, in addition to spatial operations like
rotations and mirrors, also the time-reversal operation T, which reverses the current
and the magnetic field. Note that the symmetry operations form a group, so one
can also compose them like m'xy = Tmxy.
(b) Find the magnitude of the exact magnetic field |B| at position (x, y, z) = (0, 0, 0.4) m.
Use units of 10^-6 T = mu T and state the result to three significant figures.
(c) Find the magnitude of the magnetic field within the dipole approximation |Bdip|
at the same position (x, y, z) = (0, 0, 0.4) m. Use units of 10^-6 T = mu T and state
the result to three significant figures.

Added by Zachary R.

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