Difference between revisions of "Functions composed of Physical Expressions"

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(→‎Lorentz Force for (q,q'))
(→‎Lorentz Force for (q,q'))
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A charge <math>q</math> which has a velocity of <math>\mathbf{v}</math> at <math>\left(\mathbf{r},t\right)</math> will experience a Lorentz force due to a point charge <math>q'</math> at <math>\left(\mathbf{r'},t'\right)</math> of:
 
A charge <math>q</math> which has a velocity of <math>\mathbf{v}</math> at <math>\left(\mathbf{r},t\right)</math> will experience a Lorentz force due to a point charge <math>q'</math> at <math>\left(\mathbf{r'},t'\right)</math> of:
  
−
:<math>\mathbf{F} \quad = \quad q\left[\mathbf{E} + \mathbf{v} \times \mathbf{B}\right]</math>
+
: <math>\mathbf{F} \quad = \quad q\left[\mathbf{E} + \mathbf{v} \times \mathbf{B}\right]</math>
  
 
The electric field <math>\mathbf{E}</math> is:
 
The electric field <math>\mathbf{E}</math> is:
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The magnetic field <math>\mathbf{B}</math> is:
 
The magnetic field <math>\mathbf{B}</math> is:
  
−
:<math>\mathbf{B} \quad = \quad \nabla \times \mathbf{A}</math>
+
: <math>\mathbf{B} \quad = \quad \nabla \times \mathbf{A}</math>
  
 
The Lorentz Force can be expressed directly in terms of the potentials:
 
The Lorentz Force can be expressed directly in terms of the potentials:
  
−
:<math>\mathbf{F} \quad = \quad q\left[- \nabla \varphi - \frac{∂\mathbf{A}}{∂t} + \mathbf{v} \times \left( \nabla \times \mathbf{A} \right)\right]</math>
+
: <math>\mathbf{F} \quad = \quad q\left[- \nabla \varphi - \frac{∂\mathbf{A}}{∂t} + \mathbf{v} \times \left( \nabla \times \mathbf{A} \right)\right]</math>
  
 
Where:
 
Where:
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To restate from a previous section, the magnetic vector potential of a charge <math>q</math> experienced by a charge <math>q</math> is:
 
To restate from a previous section, the magnetic vector potential of a charge <math>q</math> experienced by a charge <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>
  
−
The partial derivative of this with respect to time <math>t</math> is:
+
Using the product rule, the partial derivative of this with respect to time <math>t</math> can be found. For example, the derivative of a product of two variables <math>x</math> and <math>y</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{d}{dt}(xy) = \dfrac{dx}{dt}y + x\dfrac{dy}{dt}</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{∂^2r'}{∂t^2} } \right]</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:
  
−
Where:
+
: <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{∂r'}{∂t}</math> = the velocity of <math>q'</math> at the retarded time <math>t'</math> according to an observer at time <math>t</math> sharing the same inertial frame as <math>q</math>.
+
 
−
* <math>\frac{∂^2r'}{∂t^2}</math> = the acceleration of <math>q'</math> at the retarded time <math>t'</math> according to an observer at time <math>t</math> sharing the same inertial frame as <math>q</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{∂^2r'}{∂t^2} } \right]</math>
−
* <math>\frac{1}{|\mathbf{r}-\mathbf{r'}|}</math> = 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> (i.e. the inverse of the norm of the vector difference between the two position vectors <math>r</math> and <math>r'</math>).
+
 
−
* <math>\frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}</math> = the partial derivative of this proximity with respect to time <math>t</math>.
+
{| class="wikitable" width=480
 +
|+ 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{∂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>
 +
|}
 +
 
 +
{| class="wikitable" width=480
 +
|+ 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{∂^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>
 +
|}
  
 
==See also==
 
==See also==

Revision as of 01:52, 15 May 2016

Functions for a point charge q′

The electric scalar potential φ at (r,t) due to a point charge q′ at (r′,t′) is:

φ(r,r′)=q′4π ϵ0constant1|r−r′|proximity

The magnetic vector potential A at (r,t) due to a point charge q′ which had a velocity ∂r′∂t at (r′,t′) is:

A(r,r′)=φ(r,r′)1c2constant∂r′∂tdislocation

A(r,r′)=μ0 q′4πconstant1|r−r′|proximity∂r′∂tdislocation

Functions for an ordered pair of point charges (q,q′)

A charge q subject to an electric scalar potential φ at (r,t) due to a point charge q′ at (r′,t′) has an electric potential energy of:

qφ(r,r′)=qq′4π ϵ0constant1|r−r′|proximity

A charge q subject to a magnetic vector potential A at (r,t) due to a point charge q′ which had a velocity ∂r′∂t at (r′,t′) has a potential momentum of:

qA(r,r′)=φ(r,r′)qc2constant∂r′∂tdislocation
qA(r,r′)=μ0 qq′4πconstant1|r−r′|proximity∂r′∂tdislocation

Lorentz Force for (q,q′)

The Lorentz Force between charges (q,q′) can be derived from the scalar potential φ and the vector potential A.

A charge q which has a velocity of v at (r,t) will experience a Lorentz force due to a point charge q′ at (r′,t′) of:

F=q[E+v×B]

The electric field E is:

E=−∇φ−∂A∂t

The magnetic field B is:

B=∇×A

The Lorentz Force can be expressed directly in terms of the potentials:

F=q[−∇φ−∂A∂t+v×(∇×A)]

Where:

  • −∇φ = negative the gradient of the scalar potential φ.
  • −∂A∂t = negative the partial derivative of the magnetic vector potential A with respect to time t.
  • v×(∇×A) = the cross product of the velocity v of the charge q and the curl of the magnetic vector potential ∇×A=B due to charge q′.

To restate from a previous section, the magnetic vector potential of a charge q experienced by a charge q is:

A(r,r′)=μ0 q′4πconstant1|r−r′|proximity∂r′∂tdislocation

Using the product rule, the partial derivative of this with respect to time t can be found. For example, the derivative of a product of two variables x and y with respect to time t is:

ddt(xy)=dxdty+xdydt.

Therefore, the partial derivative of the magnetic vector potential at r due to q′ with respect to time t is:

∂A(r,r′)∂t=μ0 q′4πconstant[∂[1|r−r′|]∂tproximity∂r′∂tdislocation+1|r−r′|proximity∂[∂r′∂t]∂tdislocation]
∂A(r,r′)∂t=μ0 q′4πconstant[∂[1|r−r′|]∂tproximity∂r′∂tdislocation+1|r−r′|proximity∂2r′∂t2dislocation]
First term in the brackets
∂[1|r−r′|]∂t
= the partial derivative, with respect to time t, of the proximity of the position r of q at time t to the position r′ of q′ at the retarded time t′. 		∂r′∂t
= According to an observer at time t: the velocity a charge q′ had at the retarded time t′=t−|r−r′|/c, when it emitted a light signal which has now reached q at position r and time t
Second term in the brackets
1|r−r′|
= the proximity of the position r of q at time t to the position r′ of q′ at the retarded time t′. 		∂2r′∂t2
= According to an observer at time t: the acceleration a charge q′ had at the retarded time t′=t−|r−r′|/c, when it emitted a light signal which has now reached q at position r and time t

See also

Site map

HQ ● Glossary ● April 2016 Presentation

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