Ch7 - Electrodynamics In electrostatics and magnetostatics,...

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Unformatted text preview: Electrodynamics In electrostatics and magnetostatics, both the E-field and B- field are time-independent. They are relevant to cases where we have a fixed charge distribution and steady current Faraday discovered that a changing magnetic flux through a circuit creates an electromotive force (emf) in the circuit The changing flux may be due to a) A moving circuit loop in a region of constant magnetic field (motional emf) b) A fixed circuit loop in a constant but moving region of magnetic field c) A fixed circuit loop in a region of time-varying magnetic field In all three cases, the emf E around a circuit, which is defined by E = C d l E are described by the equation E = dt d B - , (7.1) where B is the magnetic flux through the circuit. Therefore dt d d B C - = l E (7.2) - = - = S S d t d dt d a B a B Using Stokes theorem, a E l E d d S C = ) ( - = S S d t d a B a E ) ( (7.3) Since Eq.(7.3) is valid for any chosen surface S, we have t - = B E (7.4) Eq.(7.4) is the differential form of Faradays law , while Eq.(7.2) is the integral form of Faradays law. Note that sometimes the so-called flux rule (Eq.(7.2)) may not be applied in a straightforward manner if we are dealing with motional emf. e.g. A constant magnetic field is applied through a spinning conducting disk of radius a and angular velocity . There is no change of magnetic flux through any chosen closed circuit, but yet there is a motional emf due to the magnetic force on the charge carried by the spinning disk. The emf is given by E = dl charge) t (force/uni = l B v d ) ( = a ds B s ) ( where represents a chosen path on the disk, and s is the distance from the centre of the disk. E = 2 Ba 2 , and hence the current through the circuit is given by I = R 2 Ba R 2 = E . As another e.g. of a wrong conclusion that could be caused by a straightforward application of the flux rule, consider the following situation : Here, a slight rocking of the plates will cause a big change in the flux through the closed circuit, yet there should only be negligible emf if the motion is slow enough. e.g. A uniform, but time-varying magnetic field B (t ) fills up a circular area of empty space. The induced E-field within this region can be calculated by using Faradays law : There is a cylindrical symmetry in this problem. By this symmetry, the E-field cannot have a radial component. E(2 r) = ) )( ( B r dt d 2 - E = dt dB 2 r- i.e. E = dt dB 2 r- e.g. A line charge of density is glued to the rim of a wheel of radius b . A uniform B-field is present in a region of radius a within the wheel....
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Ch7 - Electrodynamics In electrostatics and magnetostatics,...

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