What is the last step in the derivation of the Biot Savart Law?

AI Thread Summary
The discussion centers on the derivation of the Biot-Savart Law, specifically the confusion regarding the transition from the B vector to dB in the final step. The user questions how the cross product dL2 was canceled on both sides of the equation, seeking clarification on this mathematical manipulation. There is an emphasis on understanding the fundamental concepts of the Biot-Savart Law, Ampere's Law, and the Lorentz force, suggesting that a solid grasp of these principles is necessary for the derivation. The conversation highlights the importance of distinguishing between different laws and their components in electromagnetism. Overall, a deeper understanding of these foundational laws is essential for successfully completing the derivation.
CoolDude420
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Homework Statement


Hi,

So I'm having some trouble deriving the biot savart Law. We have been given a derivation in the notes and I understand everything up until the last step which is labelled in the picture?

2cdb94b9b9.png


Equating the two,
ff817eeb61.png


How did the B vector turn into dB? It seems like he canceled the cross product dL2 part on both sides? How is that possible?

Homework Equations

The Attempt at a Solution

 

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I think you are not having the knowledge of correct terms used in different laws. Please first make sure you know what is Biot savart law, it is about what. Also first learn what is Ampere's law and what is Lorentz force?
 
Let'sthink said:
I think you are not having the knowledge of correct terms used in different laws. Please first make sure you know what is Biot savart law, it is about what. Also first learn what is Ampere's law and what is Lorentz force?

I'm trying to derive the Biot Savart Law. Lorentz force is covered in a later chapter. I realize that what I have stated up there is not in fact Amperes Law but simply a component of the Lorentz Force.
 
First from the book you just write down the three laws you are naming. Derive them later.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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