Derivation of Faraday's Law from the Lorentz Force Law

  • Thread starter Lamarr
  • Start date
  • #1
52
0
Someone asked me how Faraday's Law of Induction and Ampere's Force Law, both which form part of Maxwell's Equations, are related.

Ampere's Force Law is derived from the Lorentz Force Law. They are entirely compatible with Faraday's Law of Induction. Here's how...


The Lorentz Force Law states:

[tex]F_B=Bq \times v[/tex]

[tex]B[/tex] Magnetic flux Density

[tex]q[/tex] Magnitude of charge

[tex]v[/tex] Velocity of charge




[tex]q=ALρ_q[/tex] [tex] \frac{dq}{dt}=Ap_q. \frac{dL}{dt} [/tex]

[tex]ρ_q[/tex] Charge density

[tex]A[/tex] Cross-sectional area

[tex]L[/tex] Length



[tex]v=\frac{dL}{dt}[/tex]



[tex]∴F_B=\frac{dL}{dt} \times B.ALρ_q=Ap_q. \frac{dL}{dt}×BL[/tex]
[tex]F_B=\frac{dq}{dt}×BL[/tex]


[tex]V=\frac{dW}{dq}[/tex]

[tex]V[/tex] Potential Difference

[tex]W [/tex] Work done

[tex]x[/tex] Perpendicular displacement


[tex]W=∫F_B .dx=∫\frac{dq}{dt}×BL .dx[/tex]

[tex]W=∫BL\frac{dx}{dt} .dq[/tex]

[tex]∴V=BL\frac{dx}{dt}[/tex]



[tex]BLx=\phi[/tex]

[tex]\phi[/tex] Magnetic Flux Density


Assuming B and L to be invariant:

[tex]BL\frac{dx}{dt}=\frac{d\phi}{dt}[/tex]

[tex]∴V=\frac{d\phi}{dt}[/tex]



A very crappy derivation, but it's the best possible way to show the direct connection between the two formulas.
 
Last edited:

Answers and Replies

  • #2
52
0
Hope I haven't made any mistakes. :blushing:
 
  • #3
BruceW
Homework Helper
3,611
119
  • #4
1,024
32
Are you familiar with the fact that the emf induced comes from two parts one is the motion part which you are counting and the other is flux change(both are different).think about faraday disk and see how will you apply flux rule to count for emf.
 
  • #5
52
0
Well flux change is due to motion as well.
 

Related Threads on Derivation of Faraday's Law from the Lorentz Force Law

Replies
1
Views
2K
Replies
28
Views
13K
  • Last Post
Replies
1
Views
2K
Replies
3
Views
3K
Replies
11
Views
2K
Replies
5
Views
1K
Replies
4
Views
474
  • Last Post
Replies
2
Views
4K
Replies
1
Views
2K
Replies
14
Views
4K
Top