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chapter30 - Chapter 30 Sources of the Magnetic Field A...

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Chapter 30 Sources of the Magnetic Field
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A whole picture helps Charge q as source Current I as source Electric field E Magnetic field B Gauss’s Law Ampere’s Law Force on q in the field Force on or I in the filed r q v Ampere-Maxwell Law Faraday’s Law Summarized in Maxwell equations = r r E q F E = r r r B q F v B = + r r r r q q F E v B Lorentz force
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Math -- review = ° r r r C A B Vector cross product: θ r A r B = ° r r r C A B Vector cross product: = ° = - r r r r D B A C θ r A r B = r r r D A B ( 29 sin i = r C ABθ C Determine the direction. If Magnitude of the vector : r C = ° r r r B F v B The right-hand rule: 1. Four fingers follow the first vector. 2. Bend towards the second vector. 3. Thumb points to the resultant vector.
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Sources of Electric field, magnetic field From Coulomb's Law, a point charge dq generates electric field distance away from the source: r r πε = r 2 0 1 4 dq ˆ d r E r r r r d E dq P From Biot-Savart's Law , a point current generates magnetic field distance away from the source: r Id s r r r d B r Id s μ π = r r 0 2 4 Id ˆ d r s B r Difference: 1. A current segment , not a point charge dq , hence a vector. 2. Cross product of two vectors, is determined by the right-hand rule. r Id s r d B
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Total Magnetic Field is the field created by the current in the length segment , a vector that takes the direction of the current. To find the total field, sum up the contributions from all the current elements The integral is over the entire current distribution μ o = 4 π x 10 -7 T . m / A , is the constant μ o is called the permeability of free space d B r ˆ i = r r 2 4 o μ Id π r s r B r Id s r d s
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Example: from a Long, Straight Conductor with current I The thin, straight wire is carrying a constant current I Constructing the coordinate system and place the wire along the x-axis, and point P in the X-Y plane. Integrating over all the current elements gives ( 29 I cos I ˆ sin sin = - = - = r 2 1 1 2 4 , or 4 θ o θ o μ Bθ dθ πa μ θ θ B πa B k ( 29 ˆ ˆ ˆ sin cos i = - = r 2 π d dxθ dx θ s r k k B r
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The math part ( 29 I cos ˆ ˆ ˆ cos cos , tan , cos I cos I sin sin ˆ = = =- = = - = - = - = - = r r Q r 2 1 3 2 2
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