Difference between revisions of "Functions composed of Physical Expressions"
From S.H.O.
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* <math>\mathbf{v} \times \left( \nabla \times \mathbf{A} \right)</math> = the cross product of the velocity <math>\mathbf{v}</math> of the charge <math>q</math> and the curl of the magnetic vector potential <math>\nabla \times \mathbf{A} = \mathbf{B}</math> due to charge <math>q'</math>. | * <math>\mathbf{v} \times \left( \nabla \times \mathbf{A} \right)</math> = the cross product of the velocity <math>\mathbf{v}</math> of the charge <math>q</math> and the curl of the magnetic vector potential <math>\nabla \times \mathbf{A} = \mathbf{B}</math> due to charge <math>q'</math>. | ||
− | To restate from a previous section, the magnetic vector potential from a charge <math>q'</math> at the position <math>r</math> at <math>q</math> is: | + | To restate from a previous section, the magnetic vector potential from a charge <math>q'</math> at the position <math>\mathbf{r}</math> at <math>q</math> is: |
: <math>\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}}</math> | : <math>\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}}</math> | ||
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: <math>\frac{d}{dt}(xy) = \dfrac{dx}{dt}y + x\dfrac{dy}{dt}</math>. | : <math>\frac{d}{dt}(xy) = \dfrac{dx}{dt}y + x\dfrac{dy}{dt}</math>. | ||
− | Therefore, the partial derivative of the magnetic vector potential at <math>r</math> due to <math>q'</math> with respect to time <math>t</math> is: | + | Therefore, the partial derivative of the magnetic vector potential at <math>\mathbf{r}</math> due to <math>q'</math> with respect to time <math>t</math> is: |
: <math>\frac{∂\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right)}{∂t} \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \left[ \underset{proximity}{ \frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}} + \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{ \frac{ ∂\left[ \frac{∂\mathbf{r'}}{∂t} \right] }{∂t}} \right]</math> | : <math>\frac{∂\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right)}{∂t} \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \left[ \underset{proximity}{ \frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}} + \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{ \frac{ ∂\left[ \frac{∂\mathbf{r'}}{∂t} \right] }{∂t}} \right]</math> | ||
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|+ First term in the brackets | |+ First term in the brackets | ||
|- | |- | ||
− | |width=240 valign=top align=center| <math>\frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}</math><br>The partial derivative, with respect to time <math>t</math>, of the proximity of the position <math>r</math> of <math>q</math> at time <math>t</math> to the position <math>r'</math> of <math>q'</math> at the retarded time <math>t'</math>. | + | |width=240 valign=top align=center| <math>\frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}</math><br>The partial derivative, with respect to time <math>t</math>, of the proximity of the position <math>\mathbf{r}</math> of <math>q</math> at time <math>t</math> to the position <math>r'</math> of <math>q'</math> at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>. |
|width=240 valign=top align=center| <math>\frac{∂r'}{∂t}</math><br>According to an observer at time <math>t</math>: the velocity a charge <math>q'</math> had at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>, when it emitted a light signal which has now reached <math>q</math> at position <math>\mathbf{r}</math> and time <math>t</math> | |width=240 valign=top align=center| <math>\frac{∂r'}{∂t}</math><br>According to an observer at time <math>t</math>: the velocity a charge <math>q'</math> had at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>, when it emitted a light signal which has now reached <math>q</math> at position <math>\mathbf{r}</math> and time <math>t</math> | ||
|} | |} | ||
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|+ Second term in the brackets | |+ Second term in the brackets | ||
|- | |- | ||
− | |width=240 valign=top align=center| <math>\frac{1}{|\mathbf{r}-\mathbf{r'}|}</math><br>The proximity of the position <math>r</math> of <math>q</math> at time <math>t</math> to the position <math>r'</math> of <math>q'</math> at the retarded time <math>t'</math>. | + | |width=240 valign=top align=center| <math>\frac{1}{|\mathbf{r}-\mathbf{r'}|}</math><br>The proximity of the position <math>\mathbf{r}</math> of <math>q</math> at time <math>t</math> to the position <math>r'</math> of <math>q'</math> at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>. |
|width=240 valign=top align=center| <math>\frac{∂^2r'}{∂t^2}</math><br>According to an observer at time <math>t</math>: the acceleration a charge <math>q'</math> had at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>, when it emitted a light signal which has now reached <math>q</math> at position <math>\mathbf{r}</math> and time <math>t</math> | |width=240 valign=top align=center| <math>\frac{∂^2r'}{∂t^2}</math><br>According to an observer at time <math>t</math>: the acceleration a charge <math>q'</math> had at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>, when it emitted a light signal which has now reached <math>q</math> at position <math>\mathbf{r}</math> and time <math>t</math> | ||
|} | |} |
Revision as of 14:47, 15 May 2016
Contents
Functions for a point charge
The electric scalar potential
at due to a point charge at is:
The magnetic vector potential
at due to a point charge which had a velocity at is:
Functions for an ordered pair of point charges
A charge
subject to an electric scalar potential at due to a point charge at has an electric potential energy of:A charge
subject to a magnetic vector potential at due to a point charge which had a velocity at has a potential momentum of:Lorentz Force for
The Lorentz Force between charges
can be derived from the scalar potential and the vector potential .A charge
which has a velocity of at will experience a Lorentz force due to a point charge at of:The electric field
is:The magnetic field
is:The Lorentz Force can be expressed directly in terms of the potentials:
Where:
- = negative the gradient of the scalar potential .
- = negative the partial derivative of the magnetic vector potential with respect to time .
- = the cross product of the velocity of the charge and the curl of the magnetic vector potential due to charge .
To restate from a previous section, the magnetic vector potential from a charge
at the position at is:Using the product rule, the partial derivative of this with respect to time
can be found. For example, the derivative of a product of two variables and with respect to time is:- .
Therefore, the partial derivative of the magnetic vector potential at
due to with respect to time is: The partial derivative, with respect to time , of the proximity of the position of at time to the position of at the retarded time . |
According to an observer at time : the velocity a charge had at the retarded time , when it emitted a light signal which has now reached at position and time |
The proximity of the position of at time to the position of at the retarded time . |
According to an observer at time : the acceleration a charge had at the retarded time , when it emitted a light signal which has now reached at position and time |
See also
Site map
HQ ● Glossary ● April 2016 Presentation
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