Ampère’s Law The French physicist André–Marie Ampère (1775–1836) discovered that an electrical current I in a wire produces a magnetic field B . A special case of Ampère’s Law relates the current to the magnetic field through the equation ∮ C B ⋅ d r = μ I , where C is any closed curve through which the wire passes and μ is a physical constant. Assume that the current I is given in terms of the current density J as I = ∬ S J • n d S , where S is an oriented surface with C as a boundary. Use Stokes’ Theorem to show that an equivalent form of Ampère’s Law is ▿ × B = μ J.
Ampère’s Law The French physicist André–Marie Ampère (1775–1836) discovered that an electrical current I in a wire produces a magnetic field B . A special case of Ampère’s Law relates the current to the magnetic field through the equation ∮ C B ⋅ d r = μ I , where C is any closed curve through which the wire passes and μ is a physical constant. Assume that the current I is given in terms of the current density J as I = ∬ S J • n d S , where S is an oriented surface with C as a boundary. Use Stokes’ Theorem to show that an equivalent form of Ampère’s Law is ▿ × B = μ J.
Solution Summary: The author explains the equivalent form of Ampère's Law: the current to the magnetic field is calculated through the equation displaystyle
Ampère’s Law The French physicist André–Marie Ampère (1775–1836) discovered that an electrical current I in a wire produces a magnetic field B. A special case of Ampère’s Law relates the current to the magnetic field through the equation
∮
C
B
⋅
d
r
=
μ
I
, where C is any closed curve through which the wire passes and μ is a physical constant. Assume that the current I is given in terms of the current density J as
I
=
∬
S
J
•
n
d
S
, where S is an oriented surface with C as a boundary. Use Stokes’ Theorem to show that an equivalent form of Ampère’s Law is ▿ × B = μJ.
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