Derivation of Faraday's Law from Lorentz's

In summary, the conversation discusses two ways of deriving Faraday's law from the Lorentz force law. The first method simplifies the vectors and uses u-substitution to arrive at the equation for induced EMF, while the second method keeps the vectors and uses the fact that the quantity (\vec{v}\times\vec{B}) is a time derivative of the area formed. The latter method is ultimately more successful in deriving Faraday's law, but it is noted that the inclusion of the cross product may have been necessary. The overall goal of the conversation is to derive Faraday's law from the Lorentz force law.
  • #1
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$$\vec{F}=q\vec{v}\times\vec{B}$$

$$\frac{d\vec{F}}{dq}=\vec{v}\times\vec{B}$$

$$\int\frac{d\vec{F}}{dq} \cdot ds=\int(\frac{d\vec{s}}{dt}\times\vec{B}) \cdot ds$$

from here, I went about it two different ways:

1.) Here I assumed everything was at right angles and got rid of all the vectors and vector products

$$\varepsilon=\int \frac{ds}{dt}B ds=\int \frac{ds}{dt}B \frac{ds}{dt}dt$$By u substitution

$$u=\frac{ds}{dt}, du=dt$$
$$\varepsilon=\int B(u^2)du=\frac{Bv^3}{3}$$

where v = ds/dtThat was the first way i went about it, but i didn't feel any closer to Faraday's law.

2.) Here I left the vectors alone on the RHS; I figured since [itex]\hat{v}[/itex] and d[itex]\hat{s}[/itex] were perpendicular, the quantity ([itex]\vec{v}[/itex]s) would be a time derivative of the area formed

$$\varepsilon=\int\frac{ds}{dt}B ds=\int(\vec{v}\times\vec{B}) \cdot d\vec{s}=\dot{A}B$$

$$\varepsilon=\frac{BA}{dt}$$

don't know where the minus sign is; probably was supposed to do something with the cross product, but didn't know what.Well I got a lot further with the second "method," but is this a valid derivation? and what went wrong with the first method?
 
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  • #2
Oh! PS, well, more like pre-script... my goal is to derive faraday's induction law from lorentz force law
 

1. How is Faraday's Law derived from Lorentz's Law?

Faraday's Law is derived from Lorentz's Law through the application of Maxwell's equations, specifically the Maxwell-Faraday equation. This equation states that the curl of the electric field is equal to the negative time derivative of the magnetic field. By applying this equation to a moving charged particle, we can derive Faraday's Law.

2. What is Lorentz's Law?

Lorentz's Law describes the force experienced by a charged particle moving through an electromagnetic field. It states that the force is equal to the charge of the particle multiplied by the sum of the electric field and the cross product of the velocity and magnetic field.

3. Why is Faraday's Law important in electromagnetism?

Faraday's Law is important because it describes the relationship between a changing magnetic field and the resulting induced electric field. This is crucial for understanding electromagnetic induction and is the basis for many modern technologies, such as generators and transformers.

4. Is Faraday's Law a fundamental law of physics?

Yes, Faraday's Law is considered a fundamental law of physics because it is one of Maxwell's equations, which describe the fundamental workings of electromagnetism. These equations are integral to our understanding of electricity and magnetism, and have been verified by numerous experiments.

5. How is Faraday's Law used in practical applications?

Faraday's Law is used in many practical applications, including generators, transformers, and electric motors. It is also used in wireless charging technology and electromagnetic braking systems. Additionally, Faraday's Law is the basis for techniques such as magnetic resonance imaging (MRI) and electromagnetic flow meters.

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