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ALL EXAMPLES FROM FIELDS NOTES

# ALL EXAMPLES FROM FIELDS NOTES - Notes 3 Fall 2004 Now to...

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Notes 3 Fall 2004.doc Now, to obtain the magnetic flux, Φ , through the whole surface, S, we perform a surface integral: Φ = Φ S d or in the present example, ( ) ∫∫ = Φ y , z dydz x ˆ B G . Example: In the figure below a rectangular surface is shown edge-on. Let a time-independent and spatially constant magnetic flux density be represented by 2 m Wb y ˆ 3 x ˆ 2 + = B G and let . Calculate the flux through the rectangular surface. 4 z 0 , 3 y 0 < < < < y ˆ 3 x ˆ B G θ = Φ S S d G G B dydz x ˆ S d choose and dydz dS that Note + = = G = = Φ = Φ S S S dydz x ˆ S d d B B G G G Since here B G does not change with position (it is not a function of y or z), we can pull the quantity x ˆ B G out of the integral, i.e., ( ) ( ) dydz x ˆ y ˆ 3 x ˆ 2 S d x ˆ 3 0 4 0 S + = = Φ G G B ( ) ( )( ) ( )( )( ) Wb 24 4 3 2 z 3 2 dz y 2 4 0 3 0 4 0 = = = = Φ Note that when B G is independent of position (we sometimes say spatially constant) we simply multiply x ˆ B G by the area. Often, as in the case of a circle or rectangle, we can simply substitute the area formula (that we all know, right!?) in place of the integral. Here, we could have simply replaced with . Otherwise we must perform the integration to calculate the area. So look for simplifcations of this sort. S S d G ( )( ) 12 3 4 =

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Notes 3 Fall 2004.doc Example: A single circular wire loop of radius cm 1 a = is located in the x-y plane and is centered at the origin. It is immersed in a magnetic field given by ( ) t sin z ˆ 3 y ˆ 2 ω + = o B B G (this field is spatially constant, but not temporally constant). For T 05 . 0 = o B and an angular frequency of s rad 377 , calculate the magnetic flux through the surface, Φ , at ms 77 . 2 t = . y Let us choose and note that dS z ˆ S d = G 2 S a z ˆ S z ˆ S d π = = G . Then x ( ) ( ) ( ) ( ) ( ) ( ) Wb 7 . 40 01 . 0 10 77 . 2 377 sin 05 . 0 3 a t sin 3 S t sin 3 dS t sin 3 dS t sin 3 dS z ˆ t sin z ˆ 3 y ˆ 2 S d 2 3 2 S S S S µ = π × = π ω = ω = ω = ω = ω + = = Φ o o o o o B B B B B B G G Example: A rectangular surface (seen edge-on below) is located in the x-y plane and has extent . It is immersed in a magnetic field that is given by cm 5 y 0 , cm 3 x 0 < < < < ( ) t sin z ˆ y 3 y ˆ 2 ω + = o B B G . This field is not spatially (nor temporally) constant since it is a function of y. For and an angular frequency of T 04 . 0 = o B s rad 377 , calculate the magnetic flux, Φ , at ms 77 . 2 t = . ( ) ω + = = Φ 03 . 0 0 05 . 0 0 S dxdy z ˆ t sin z ˆ y 3 y ˆ 2 S d o B B G G 4 y z B G 5 ( ) ω = Φ 03 . 0 0 05 . 0 0 dxdy t sin y 3 o B dxdy z ˆ dS z ˆ S d choose = = G ydy x t sin 3 05 . 0 0 03 . 0 0 ω = Φ o B ( ) ω = Φ 05 . 0 0 2 2 1 y 03 . 0 t sin 3 o B ( )( ) ( ) ( ) ( ) ( ) 3 4 2 2 1 10 77 . 2 377 sin 10 125 . 1 t sin 05 . 0 03 . 0 04 . 0 3 × × = ω = Φ Wb 9 . 3 µ = Φ