Earth’s magnetic field A cross section of Earth’s magnetic field can be represented as a vector field in which the center of Earth is located at the origin and the positive y-axis points in the direction of the magnetic north pole. The equation for this field is F ( x , y ) = M ( x , y ) i + N ( x , y ) j = m ( x 2 + y 2 ) 5 / 2 [ 3 x y i + ( 2 y 2 − x 2 ) j ] where m is the magnetic moment of Earth. Show that this vector field is conservative.
Earth’s magnetic field A cross section of Earth’s magnetic field can be represented as a vector field in which the center of Earth is located at the origin and the positive y-axis points in the direction of the magnetic north pole. The equation for this field is F ( x , y ) = M ( x , y ) i + N ( x , y ) j = m ( x 2 + y 2 ) 5 / 2 [ 3 x y i + ( 2 y 2 − x 2 ) j ] where m is the magnetic moment of Earth. Show that this vector field is conservative.
Solution Summary: The author explains that the Earth's magnetic field m(x2+y2)fric52, is conservative.
A cross section of Earth’s magnetic field can be represented as a vector field in which the center of Earth is located at the origin and the positive y-axis points in the direction of the magnetic north pole. The equation for this field is
F
(
x
,
y
)
=
M
(
x
,
y
)
i
+
N
(
x
,
y
)
j
=
m
(
x
2
+
y
2
)
5
/
2
[
3
x
y
i
+
(
2
y
2
−
x
2
)
j
]
where m is the magnetic moment of Earth. Show that this vector field is conservative.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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