1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Flux through rim created by moving circuit and induced EMF

  1. Jan 8, 2017 #1
    In Purcell- Electricty and Magnetism book, in the chapter on electromagnetic induction, I found the following explanation regarding the magnetic flux through a circuit of area ##S##.

    Consider the circuit in figure, moving in a time ##dt## in a magnetic field ##B##, constant in time (but not uniform in space).
    6666666666666666666666666.png
    The flux through the surface of the circuit ##S## at time ##t## is
    $$\Phi(t)=\int_{S} B \cdot da$$
    And I'm totally ok with that. Nevertheless it is said that the flux at time ##t+dt## is
    $$\Phi(t+dt)=\int_{S+dS} B \cdot da=\Phi(t) +\int_{dS} B \cdot da$$
    i.e. ##\Phi(t+dt)## is given by the previous flux, plus the flux through the "rim" ##dS## (let's call this ##\Phi_{dS}##).
    Here is the problem: I think that stating
    $$-\Phi(t+dt)+\Phi(t)+\Phi_{dS}=0$$
    is against the law ##\nabla \cdot B=0##. Infact, considering the volume enclosed in the circuit surface at time ##t## and ##t+dt## togheter with the surface of the "rim", then ##\Phi(t+dt)## and ##\Phi_{dS}## are outgoing fluxes from the volume, while ##\Phi(t)## is an ingoing flux in the volume.
    Therefore, if the previous relation is true, than the total flux outgoing from the volume would be
    $$+\Phi(t+dt)-\Phi(t)+\Phi_{dS}= 2 \Phi_{dS} \neq 0$$
    Is this correct or is there something missing?

    In the book, from this proof it is derived that the emf induced in the circuit is
    $$\mathrm{emf}=-\frac{d \Phi}{dt}=-\frac{d \Phi_{dS}}{dt} \tag{1}$$
    (indicating with ##\Phi## the flux through the circuit surface ##S## and with ##\Phi_{dS}## the flux through the rim).
    But, as I also see on another textbook, I think it should be
    $$\mathrm{emf}=-\frac{d \Phi}{dt}=+\frac{d \Phi_{dS}}{dt} \tag{2}$$
    Which of the two is the correct one, ##(1)## or ##(2)##?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Flux through rim created by moving circuit and induced EMF
Loading...