ECE-305-Lecture17 - Engineering Electro-Magnetics...

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Engineering Electro-Magnetics Engineering Electro-Magnetics ECE-305, Lecture 17 Dennis McCaughey, Ph.D. 25 March 2010
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03/25/2010 Dennis McCaughey, ECE305,Spring 2010 2 HWK HWK 9.6 Due 4/1
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03/25/2010 Dennis McCaughey, ECE305,Spring 2010 3 Scalar Magnetic Potential Scalar Magnetic Potential Can we define a potential function which may be found from the current distribution and from which the magnetic fields may be easily determined? Yes Can a scalar magnetic potential be defined? “Sometimes” ( 29 Assume a scalar magnetic potential exists This definition must not conflict with our previous results Curl of the gradient of a scalar is identically equal to zero. Therefor if is to m m V V = -∇ ∇× = = ∇× -∇ H H J H ( 29 be defined as the gradient of a scalar magnetic potential, the current density must be zero throughout the region in which the scalar magnetic potential is to be defined. 0 m V = -∇ = H J
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03/25/2010 Dennis McCaughey, ECE305,Spring 2010 4 Continued Continued Scalar magnetic potential can be useful since many problems involve geometries in which the current- carrying conductors occupy a relatively small fraction of the total region of interest. It is also applicable to permanent magnets ( 29 ( 29 0 0 2 The scalar magnetic potential satisfies Laplace's equation. In free space 0 0 0 0 It is not defined in any region in which a current density exists One difference between the ele m m V V μ μ ∇• = ∇ • = ∇ • -∇ = = = B H J ctric potential and the scalar magnetic potential is the while is single valued is not m V V
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03/25/2010 Dennis McCaughey, ECE305,Spring 2010 5 Scalar Magnetic Potential is not a Single Valued Function Scalar Magnetic Potential is not a Single Valued Function 0 in the region 2 is the current carried by the inner conductor in the direction Thus we can define a scalar magnetic potential 1 2 2 Thus 2 z m m m m m a b I a I a V I V V V I I V φ φ ρ πρ πρ ρ φ φ π φ π = < < = = -∇ = -∇ = - = - = - J H H
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03/25/2010 Dennis McCaughey, ECE305,Spring 2010 6 Scalar Magnetic Potential is not a Single Valued Function Scalar Magnetic Potential is not a Single Valued Function What value do we associate with the point P, where 4? If we let be zero at 0 and proceed counterclockwise around the circle, the magnetic potential goes negative linearly. With one circle, m V V φ π φ = = ( 29 ( 29 ( 29 but we let be zero at 0 Thus at , 4, 9 4, 17 4 2 1 2 1 0 1, 2, 3, 2 2 Why the difference with the electrostatic potential? 0 and 0 because is a conservative field m m mP I V P I I V n n n d E φ φ π π π π π π = - = = = - = - = ± ± ± ∇× = = E E L L K Ñ ( 29 and therefore the integral is independent of the path In the magnetic case, however 0 wherever 0 , but even if is zero along the path of integration. Every time we make an
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