Understanding the math of definition of electromotive force

AI Thread Summary
The discussion centers on the relationship between electromotive force (emf) and electric and magnetic fields. The equation proposed, which combines both electric field and magnetic force terms, is not commonly found in standard texts. Two scenarios are examined: one where a current loop moves in a uniform magnetic field, generating emf through magnetic force, and another where the magnetic field changes over time, inducing emf according to Faraday's law. The key question is whether these scenarios can be unified under the proposed equation, suggesting that both cases are special instances of a broader relationship. The combination of a moving loop in a time-varying magnetic field would necessitate the use of the comprehensive equation to accurately describe the induced emf.
zenterix
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Homework Statement
Force on an electric charge is given by

$$\vec{F}=q(\vec{E}+\vec{v}\times\vec{B})\tag{1}$$
Relevant Equations
Emf is defined as the integral of the force per unit charge around a current loop.

$$\mathcal{E}=\oint_C\frac{d\vec{F}}{dq}\cdot d\vec{r}\tag{2}$$
Does this mean we can write the following?

$$\mathcal{E}=\oint_C \vec{E}\cdot d\vec{r}+\oint_C \vec{v}\times\vec{B}\cdot d\vec{r}\tag{3}$$

I haven't seen an equation like the above in my books and notes yet.

What I have seen are two cases.

In one case, we have a uniform magnetic field and we move the current loop in the field in such a way that we have changing magnetic flux.

A simple example of this is a rectangular current loop where one of the sides is movable. This side has velocity ##\vec{v}## and so experiences the effect of a magnetic force ##\vec{v}\times\vec{B}##.

$$\mathcal{E}=\oint_C \vec{v}\times\vec{B}\cdot d\vec{r}\tag{4}$$

Is the ##\vec{E}## term zero in (1) for this case?

In a second case, the conducting loop does not move but the magnetic field is allowed to change in time.

We get an induced emf in the loop but we can't explain it with the Lorentz force.

We explain it with a new law of nature, Faraday's law, which says that

$$\nabla\times \vec{E}=-\frac{\partial\vec{B}}{\partial t}\tag{5}$$

If we integrate both sides along a surface ##S##, then by Stokes' theorem the lhs side is

$$\oint_C \vec{E}\cdot d\vec{s}=\iint_S(\nabla\times \vec{E})\cdot d\vec{a}\tag{6}$$

and so

$$\oint_{C} \vec{E}\cdot d\vec{s}=\iint_S (\nabla\times\vec{E})\cdot\hat{n}da=-\iint_S\frac{\partial \vec{B}}{\partial t}\cdot \hat{n}da\tag{7}$$

Note that the first equality is from Stokes' theorem and the second is from Faraday's law.

We can rewrite as

$$\oint_C \vec{E}\cdot d\vec{s}=-\frac{\partial}{\partial t}\iint_S \vec{B}\cdot \hat{n}da=-\frac{d\Phi_B}{dt}\tag{8}$$

and so

$$\mathcal{E}=-\frac{\partial\Phi_B}{dt}\tag{9}$$

So my question is about comparing the equations

$$\mathcal{E}=\oint_C \vec{v}\times\vec{B}\cdot d\vec{r}$$

$$\mathcal{E}=\oint_C \vec{E}\cdot d\vec{r}$$

Are these just special cases of (3)?
 
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zenterix said:
In one case, we have a uniform magnetic field and we move the current loop in the field in such a way that we have changing magnetic flux.A simple example of this is a rectangular current loop where one of the sides is movable.

zenterix said:
In a second case, the conducting loop does not move but the magnetic field is allowed to change in time.

Combination of these,i.e. we have time changing magnetic field and we have a rectangular current loop where one of the sides is movable, would requires the formula (3) to explain its emf.
 
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