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A Practical Guide to 'Free-Energy' Devices

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or in another form:<br />

So, the e.m.f. around a closed path consists in general of two components. The component dφ/dt is the<br />

variational e.m.f. and the second component is the motional E-field. In equation (9), (V x B)dl can, by<br />

means of a vec<strong>to</strong>r identity, be replaced with B x (V x dl)A. V is the sideways velocity of d: the vec<strong>to</strong>r V x dl<br />

has magnitude Vdl and a direction normal <strong>to</strong> the surface ds swept out by the moving length dl in time dt.<br />

Letting Bn denote the component of B normal <strong>to</strong> this area, we can see that the quantity -B x (V x dl)<br />

becomes -BnVdl and equation 9 can be re-written as:<br />

Clearly, the integral of BnV around the closed con<strong>to</strong>ur C with sideways velocity of magnitude V for each<br />

length dl traversed, is simply the time rate of change of the magnetic flux through the surface bounded by<br />

C. This change is directly due <strong>to</strong> the passage of path C through lines of magnetic flux. Hence, the<br />

complete expression for e.m.f. in equation (10) is the time rate of change of the magnetic flux over any<br />

surface S, bounded by the closed path C, due <strong>to</strong> the changing magnetic field and the movement of the<br />

path through the magnetic field. Equation (10) may be written:<br />

Note: The distinction between equations (7) and (11) is that equation (7) contains only the variational<br />

e.m.f. while equation (11) is the sum of the variational and motional e.m.f. values. In equation (7), the<br />

partial time derivative of magnetic flux φ is the rate of flux change due only <strong>to</strong> the time-varying magnetic<br />

field, while equation (11) includes the <strong>to</strong>tal time derivative of the rate of flux change due <strong>to</strong> the timevarying<br />

magnetic field and path C's passage through the magnetic field. If the closed path C is not<br />

passing through lines of magnetic flux, then equation (7) and equation (11) are equivalent.<br />

It is also important <strong>to</strong> point out that dφ/dt in equation (11) does not necessarily mean the <strong>to</strong>tal time rate of<br />

change of the flux φ over the surface S. For example, the flux over surface S is bounded by the closed<br />

con<strong>to</strong>ur C of the left portion of the electric circuit shown in Fig.1.<br />

The flux is changing as the coil is unwound by the rotation of the cylinder, as illustrated. However, since<br />

B is static, there is no variational e.m.f. and since the conduc<strong>to</strong>rs are not modulating lines of flux, there is<br />

A - 1207

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