44c44
< {(Version March 11, 2011)} \vspace{2.0cm} \vfill
---
> {(Version March 6, 2011)} \vspace{2.0cm} \vfill
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< {\it inertial frames}. For a freely moving body, i.e., a body which is not 
---
> {\it inertial frames}. For a freely moving body, i.e. a body which is not 
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< The covariant metric tensor lowers the indices, i.e., transforms
---
> The covariant metric tensor lowers the indices, i.e. transforms
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< is proportional to the energy, i.e., the zero component of the 
---
> is proportional to the energy, {\it i.e.} the zero component of the 
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< we have suppressed the space-time dependence, i.e.,
---
> we have suppressed the space-time dependence, {\it i.e.}
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< for free. The dual electromagnetic tensor is defined by
---
> for free. The dual electromagnetic tensor is defined
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<   {^*F}^{\alpha\beta} = {1\over 2} \epsilon^{\alpha\beta\gamma\delta}
<                         F_{\gamma\delta} , \label{DFab}
---
> {^*F}^{\alpha\beta} = {1\over 2} \epsilon^{\alpha\beta\gamma\delta}
>                           F_{\gamma\delta} , \label{DFab}
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<   \partial_{\alpha} {^*F}^{\alpha\beta} = 0 . \label{HMEQ}
---
> \partial_{\alpha} {^*F}^{\alpha\beta} = 0 . \label{HMEQ}
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< An equivalent way to write (\ref{HMEQ}) is the equation
---
> An equivalent way to write it is the equation
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<     \partial^{\alpha}F^{\beta\gamma} + \partial^{\beta}F^{\gamma\alpha} 
<   + \partial^{\gamma} F^{\alpha\beta} = 0\ .
---
> \partial^{\alpha} F^{\beta\gamma} + \partial^{\beta} F^{\gamma\alpha} 
> + \partial^{\gamma} F^{\alpha\beta} = 0\ .
1449a1450
> 
1470c1471,1475
< which is of course passed. 
---
> which is of course passed. It may be noted that, in contrast to the 
> inhomogeneous equations, the homogeneous equations determine the 
> relations with the $\vec{E}$ and $\vec{B}$ fields only up to an 
> over-all $\pm$ sign, because there is no current on the 
> right-hand side.
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< choice of a convenient gauge is at the heart of many calculations.
---
> choice of a convenient gauge is at the heart of many application.
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< transformations of second kind. Gauge transformations of first kind 
< transform fields by a constant phase, whereas for gauge transformation 
< of the second kind a space--time dependent function is encountered.} 
---
> transformations of 2.~kind. Gauge transformations of 1.~kind transform 
> fields by a constant phase, whereas for gauge transformation of the 
> 2.~kind a space--time dependent function is encountered.} 
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< As for any Lorentz tensor, we immediately know the behavior of 
< $(F^{\alpha\beta})$ under Lorentz transformations
---
> As for any Lorentz tensor, we immediately know its behavior under
> Lorentz transformation
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<   F'^{\alpha\beta} = a^{\alpha}_{\ \gamma}\,a^{\beta}_{\ \delta}\,
<   F^{\gamma\delta}\ .
---
> F'^{\alpha\beta} = a^{\alpha}_{\ \gamma}\,a^{\beta}_{\ \delta}\,
> F^{\gamma\delta}\ .
1504,1505c1509,1510
< to $\vec{E}$ and $\vec{B}$ fields, it is left as a straightforward 
< exercise to derive the transformation laws
---
> to $\vec{E}$ and $\vec{B}$ fields, it is left as an exercise for the 
> reader to derive the transformation laws
1524,1527c1529,1534
< of a charged particle. On a deeper level this phenomenon is related 
< to the conservation of energy and momentum and the fact that an
< electromagnetic field carries energy as well as momentum. Here we 
< are content with finding the Lorentz covariant force.
---
> of a charged particle. On a deeper level this phenomenon is related to
> the conservation of energy and momentum and the fact that an
> electromagnetic carries energy as well as momentum. Here we are 
> content with finding the 
> Lorentz covariant form, assuming we know already that such the 
> approximate relationship.
1546c1553
<   dp^{\alpha} = \pm {q\over c}\, F^{\alpha\beta}\, dx_{\beta}\ .
---
>  dp^{\alpha} = \pm {q\over c}\, F^{\alpha\beta}\, dx_{\beta} .
1549,1551c1556,1558
< conservation, which is in this context known as {\it Lenz law}, 
< that the force between charges of equal sign has to be repulsive. 
< This corresponds to the plus sign and we arrive at
---
> conservation, in this context known as Lenz's law, that the force 
> between charges of equal sign has to be repulsive. This corresponds 
> to the plus sign and we arrive at
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<   dp^{\alpha} = {q\over c}\, F^{\alpha\beta}\, dx_{\beta}\ .
---
>   dp^{\alpha} = {q\over c}\, F^{\alpha\beta}\, dx_{\beta} .
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< i.e., to use the 4-momentum $p^{\alpha}$
---
> {\it i.e.} to use the 4-momentum $p^{\alpha}$
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< formulation, which includes Dirac's equation for electrons and leads 
< to Quantum Electrodynamics.
---
> formulation, which ultimately has to include Dirac's equation for 
> electrons and leads then to Quantum Electrodynamics.
1588,1589c1595,1596
< To get the space component of the Lorentz force we use (\ref{FEB}) 
< and (\ref{EMfields}) and get the equality
---
> To get the space component of the Lorentz force we use besides
> (\ref{FEB}) equation (\ref{EMfields}) which give the equality
1592c1599
< The space components combine into the well-known {\it Lorentz force}
---
> The space components combine into the well-known equation
1595c1602
< 	+ {q\over c}\, \vec{U} \times \vec{B}\,,
---
> 	+ {q\over c}\, \vec{U} \times \vec{B}
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< which reveals that the relativistic velocity
---
> where our derivation reveals that the relativistic velocity
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< The equation (\ref{LFspace}) for $\vec{f}$ can be used to define 
< a measurement prescription for an electric charge unit.
< % \hfill\break % \vfill\eject
---
> The equation (\ref{LFspace}) for $\vec{f}$ may now be used to 
> define a measurement prescription for an electric charge unit.
> % \hfill\break
> % \vfill\eject
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< /\partial t)\,(\partial x^1/\partial x_1) = 0$), the vector identity
---
> /\partial t)\,(\partial x^1/\partial x_1) = 0$), the well-known
> vector identity
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< and we transform the last integral in (\ref{F1}) as follows
---
> and we transform the last integral in (\ref{F1}) as follows:
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<   &=& - \oint_C \left(\vec{\beta}\times\vec{B}\right)\cdot d\vec{l}\,,
---
>   &=& - \oint_C \left(\vec{\beta}\times\vec{B}\right)\cdot d\vec{l} 
1659c1668
<   \cdot d\vec{l} = - {1\over c} {d\ \over dt} \Phi_m\,,
---
>   \cdot d\vec{l} = - {1\over c} {d\ \over dt} \Phi_m
1675c1684
<   = - {1\over c} {d\ \over dt} \Phi_m\,,
---
>   = - {1\over c} {d\ \over dt} \Phi_m
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< is stated in most test books. Due to our initial treatment of special 
---
> is found in most test books. Due to our initial treatment of special 
1685c1694
< \subsection{Lenz's Law\label{Lenz_law}}
---
> \subsection{Lenz's law\label{Lenz_law}}
1689c1698
< This is known as \index{Lenz law}{\it Lenz law}. For closed, 
---
> This is known as \index{Lenz's law}{\it Lenz's law}. For closed, 
1691,1693c1700,1702
< whose magnitude depends on the resistance of the circuit. Lenz's law 
< states: {\it The induced emf and induced current are in such a 
< direction as to oppose the change that produces them.} Tipler 
---
> whose magnitude depends on the resistance of the circuit. \index{Lenz's 
> law}Lenz's law states: {\it The induced emf and induced current are in 
> such a direction as to oppose the change that produces them.} 
1701,1712c1710,1722
< and move the magnet toward the loop. The magnetic field through the 
< loop gets stronger when the magnet is approaching and a current is 
< induced in the loop. The direction of the current is such that its 
< magnetic field is opposite to that of the magnet. Effectively the 
< loop becomes a magnet with north pole towards the bar magnet. The 
< result is a {\it repulsive} force between bar magnet and loop. Work 
< against this force is responsible for the induced current and its 
< associated heat in the loop. Would the sign of the induced current 
< be different, an attractive force would result and the resulting 
< acceleration of the bar magnet as well as the heat in the loop 
< would violate energy conservation. Note that pulling the bar magnet 
< out of the loop does also produce energy.
---
> and move the magnet toward the loop. The magnet's magnetic field 
> through the loop get stronger when the magnet is approaching and
> a current is induced in the loop. The direction of the current is
> such that its magnetic field is opposite to that of the magnet,
> effectively the loop becomes a magnet with north pole towards the
> bar magnet. The result is a {\it repulsive} force between bar
> magnet and loop. Work against this force is responsible for the
> induced current, and its associated heat, in the loop. Would the
> sign of the induced current be different an attractive force would
> result and the resulting acceleration of the bar magnet as well
> as the heat in the loop would violate energy conservation. Note 
> that pulling the bar magnet out of the loop does also produce 
> energy.