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Exercise:
The magnetic flux through a loop of wire is given by the function Phi_mt. Determine a formal expression for the induced emf as a function of time resulting from the fluxes in a to c and find the units of the respective parameters. Graph the flux and the induced emf. abcliste abc Phi_mt Phi_left-leftfractt_-right^right abc Phi_mt Phi_sinomega t abc Phi_mt Phi_ e^-mut-t_^ abcliste

Solution:
abcliste abc mathcalEt -dotPhi_mt -Phi_ dvleft-leftfractt_-right^rightt +Phi_ leftfractt_-rightdvleftfractt_-rightt fracPhi_t_leftfractt_-right The units are: itemize item peak flux Phi_: Phi_uPhi item peak time t_: t_utp itemize includegraphicswidthcm#image_path:faraday-law-parabola-# abc mathcalEt -dotPhi_mt -Phi_ dvsinomega tt -Phi_ cosomega tdvomega tt -Phi_omegacosomega t The units are: itemize item peak flux Phi_: Phi_uPhi item angular frequency omega: omegauom itemize includegraphicswidthcm#image_path:faraday-law-sinusoidal-# abc mathcalEt -dotPhi_mt -Phi_ dvlefte^-mut-t_^rightt -Phi_ e^-mut-t_^ dvleft-mut-t_^rightt +mut-t_Phi_ e^-mut-t_^ dvt-t_t muPhi_ t-t_ e^-mut-t_^ The induced emf has extremal pos for dotmathcalEt: dotmathcalEt mu Phi_leftdvt-t_t e^-mut-t_^+t-t_dvlefte^-mut-t_^righttright muPhi_lefte^-mut-t_^+t-t_ e^-mut-t_^ left-mut-t_rightright muPhi_ e^-mut-t_^left-mut-t_^right This expression is equal to for t-t_ pmfracsqrtmu The extrema are therefore mathcalEpmfracsqrtmu pm muPhi_fracsqrtmu e^-mufracmu pmPhi_sqrtmu e^-frac pmPhi_sqrtfracmue The units are: itemize item peak flux Phi_: Phi_uPhi item peak time t_: t_utp item width parameter mu: muumu itemize includegraphicswidthcm#image_path:faraday-law-gauss# abcliste
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Exercise:
The magnetic flux through a loop of wire is given by the function Phi_mt. Determine a formal expression for the induced emf as a function of time resulting from the fluxes in a to c and find the units of the respective parameters. Graph the flux and the induced emf. abcliste abc Phi_mt Phi_left-leftfractt_-right^right abc Phi_mt Phi_sinomega t abc Phi_mt Phi_ e^-mut-t_^ abcliste

Solution:
abcliste abc mathcalEt -dotPhi_mt -Phi_ dvleft-leftfractt_-right^rightt +Phi_ leftfractt_-rightdvleftfractt_-rightt fracPhi_t_leftfractt_-right The units are: itemize item peak flux Phi_: Phi_uPhi item peak time t_: t_utp itemize includegraphicswidthcm#image_path:faraday-law-parabola-# abc mathcalEt -dotPhi_mt -Phi_ dvsinomega tt -Phi_ cosomega tdvomega tt -Phi_omegacosomega t The units are: itemize item peak flux Phi_: Phi_uPhi item angular frequency omega: omegauom itemize includegraphicswidthcm#image_path:faraday-law-sinusoidal-# abc mathcalEt -dotPhi_mt -Phi_ dvlefte^-mut-t_^rightt -Phi_ e^-mut-t_^ dvleft-mut-t_^rightt +mut-t_Phi_ e^-mut-t_^ dvt-t_t muPhi_ t-t_ e^-mut-t_^ The induced emf has extremal pos for dotmathcalEt: dotmathcalEt mu Phi_leftdvt-t_t e^-mut-t_^+t-t_dvlefte^-mut-t_^righttright muPhi_lefte^-mut-t_^+t-t_ e^-mut-t_^ left-mut-t_rightright muPhi_ e^-mut-t_^left-mut-t_^right This expression is equal to for t-t_ pmfracsqrtmu The extrema are therefore mathcalEpmfracsqrtmu pm muPhi_fracsqrtmu e^-mufracmu pmPhi_sqrtmu e^-frac pmPhi_sqrtfracmue The units are: itemize item peak flux Phi_: Phi_uPhi item peak time t_: t_utp item width parameter mu: muumu itemize includegraphicswidthcm#image_path:faraday-law-gauss# abcliste
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Attributes & Decorations
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Magnetism
Tags
faraday law, induced emf
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(4, default)
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Language
ENG (English)
Type
Calculative / Quantity
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