Notes 3 Fall 2004.doc Now, to obtain the magnetic flux, Φ, through the whole surface, S, we perform a surface integral: ∫Φ=ΦSdor in the present example, ()∫∫⋅=Φy,zdydzxˆBG. 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 2mWbyˆ3xˆ2+=BGand let . Calculate the flux through the rectangular surface. 4z0,3y0<<<<yˆ3xˆBGθ∫⋅=ΦSSdGGBdydzxˆSdchooseanddydzdSthatNote+==G∫∫∫⋅=⋅=Φ=ΦSSSdydzxˆSddBBGGGSince here BGdoes not change with position (it is not a function of y or z), we can pull the quantity xˆ⋅BGout of the integral, i.e., ()()dydzxˆyˆ3xˆ2Sdxˆ3040S∫∫∫⋅+=⋅=ΦGGB()( )( )( )( )( )Wb24432z32dzy2403040====Φ∫Note that when BGis independent of position (we sometimes say spatially constant) we simply multiply xˆ⋅BGby 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. ∫SSdG( )( )1234=
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Notes 3 Fall 2004.doc Example: A single circular wire loop of radius cm1a=is located in the x-y plane and is centered at the origin. It is immersed in a magnetic field given by ()tsinzˆ3yˆ2ω+=oBBG(this field is spatially constant, but not temporally constant). For T05.0=oBand an angular frequency of srad377, calculate the magnetic flux through the surface, Φ, at ms77.2t=. y Let us chooseand note thatdSzˆSd=G2SazˆSzˆSdπ==∫G. Then x ()()( )()()()Wb7.4001.01077.2377sin05.03atsin3Stsin3dStsin3dStsin3dSzˆtsinzˆ3yˆ2Sd232SSSSµ=⋅π×⋅=πω=ω=ω=ω=⋅ω+=⋅=Φ−∫∫∫∫oooooBBBBBBGGExample: 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 cm5y0,cm3x0<<<<()tsinzˆy3yˆ2ω+=oBBG. This field is notspatially (nor temporally) constant since it is a function of y. For and an angular frequency of T04.0=oBsrad377, calculate the magnetic flux, Φ, at ms77.2t=. ()∫∫∫⋅ω+=⋅=Φ03.0005.00Sdxdyzˆtsinzˆy3yˆ2SdoBBGG4y z BG5 ()∫∫ω=Φ03.0005.00dxdytsiny3oBdxdyzˆdSzˆSdchoose==Gydyxtsin305.0003.00∫⎟⎠⎞⎜⎝⎛ω=ΦoB()⎟⎟⎠⎞⎜⎜⎝⎛ω=Φ05.00221y03.0tsin3oB()()( )()()()342211077.2377sin10125.1tsin05.003.004.03−−×⋅×=ω=ΦWb9.3µ=Φ