Ch7 - Electrodynamics • In electrostatics and...

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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 Φ- , (7.1) where Φ B is the magnetic flux through the circuit. Therefore dt d d B C Φ- = ⋅ ∫ l E (7.2) ∫ ∫ ⋅ ∂ ∂- = ⋅- = S S d t d dt d a B a B Using Stoke’s 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 Faraday’s law , while Eq.(7.2) is the integral form of Faraday’s 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 ϖ . 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 = ∫ ⋅ × Γ l B v d ) ( = ∫ a ds B s ) ( ϖ where Γ represents a chosen path on the disk, and s is the distance from the centre of the disk. ∴ E = 2 Ba 2 ϖ , and hence the current through the circuit is given by I = R 2 Ba R 2 ϖ = 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 Faraday’s law : There is a cylindrical symmetry in this problem. By this symmetry, the E-field cannot have a radial component. ∴ E(2 π r) = ) )( ( B r dt d 2 π- ⇒ E = dt dB 2 r- i.e. E = φ ˆ dt dB 2 r- e.g. A line charge of density λ 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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This note was uploaded on 10/14/2010 for the course PHYS 2051 taught by Professor Drkwong during the Spring '10 term at CUHK.

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Ch7 - Electrodynamics • In electrostatics and...

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