L20_Viet_More Magnetic Effects

L20_Viet_More Magnetic Effects - Physics 122 Electricity...

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Physics 122 Electricity and Magnetism Lecture 20 More Magnetic Effects
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05/12/09 Physics 122 - Lecture 120 2 Line Integrals Made Easy If B is everywhere perpendicular to the path of integration ds , then: 0 f i B ds = r r If B is everywhere parallel to the path of integration ds , then: f i B ds BL = r r
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05/12/09 Physics 122 - Lecture 120 3 Ampere’s Law A special case of a line integral is one that runs in a closed path and returns to where it started, i.e., a line integral around a closed curve, which, for a magnetic field, is denoted by: Consider the case of the field at a distance d from a long straight wire: 0 2 4 I B d μ π = This result is: independent of the shape of the curve around the wire; independent of where the current passes through the curve; depends only on the amount of current passing through the integration path. Ampere’s Law Bds r r O Bds r r O = 2 π d B = μ 0 I Bds r r O = μ 0 I
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05/12/09 Physics 122 - Lecture 120 4 Example : The Magnetic Field Inside a Current-Carrying Wire A wire of radius R carries current I uniformly distributed across its cross section. Find the magnetic field inside the wire at a distance r<R from the axis. 2 2 through 2 2 I r I JA r I R R π = = = 2 0 through 0 2 (2 ) B ds B ds B r r I I R μ = = = = r r 2 0 0 2 2 2 2 4 I r Ir B r R R = = Bds r r O
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05/12/09 Physics 122 - Lecture 120 5 The Magnetic Field of a Solenoid (1) A solenoid is a helical coil of wire consisting of multiple loops, all carrying the same current. One can think of the field of a solenoid by superimposing the fields from several loops, as shown in the lower figure. On the axis, the three fields will add to make a stronger net field, but outside the loop the fields from loops 1 and 3 will tend to cancel the field from coil 2. When the fields from all the loops are superimposed, the result is that the field inside the solenoid is strong and roughly parallel to the axis, while the field outside is very weak. In the limit of an ideal solenoid the field inside is uniform and parallel to the axis, while the field outside is zero.
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05/12/09 Physics 122 - Lecture 120 6 The Magnetic Field of a Solenoid (2) We can use Ampere’s Law to calculate the field of an ideal long solenoid by choosing the integration path carefully. We choose a rectangular LxW loop, with one horizontal side outside the solenoid and the vertical sides passing through. If the loop encloses N wires, then I
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This note was uploaded on 04/25/2009 for the course ECE PHYS122 taught by Professor Viet during the Spring '09 term at 東京大学.

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L20_Viet_More Magnetic Effects - Physics 122 Electricity...

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