# B-Fields: Deriving Faraday's Law & Why No Work?

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• ChanceLiterature
In summary, the conversation discusses the confusion surrounding the concept of work in relation to B-fields. Work is mathematically defined as the path integral of a force, but it is also restricted to conservative fields. B-fields do not do work on point charges because the Lorentz force is always perpendicular to the charge's velocity. Additionally, the closed line integral of B is zero, as B is a conservative field with zero divergence. This is tied to Faraday's law and the concept of emf.
ChanceLiterature
This statement has always confused me. Now my confusion is coming home to roast while we cover EMF.

If we consider work mathematically as ∫f.dl and require integral to be path independent then of course the B-field does no work.

However, it seems like there is a deeper meaning to B-fields do no work. Is there?

Tied into this is faradays law. Faraday's Law can be "derived" from emf (it's in quotes because I understand faradays is first principles). In this derivation emf force is stated to be closed integral ∫E.dl. I do not understand why this why emf or this induced force around the loop is necessarily due to an E-field as opposed to to B-field.

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For some clarification I think you are saying that B-field do no work to charges. B-field do work to magnetic dipoles as we see two magnets attract and collide.
ChanceLiterature said:
In this derivation emf force is stated to be closed integral ∫E.dl.
it equals to
$$-\int_A\frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A}$$
using B.

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cabraham and ChanceLiterature
anuttarasammyak said:
For some clarification I think you are saying that B-field do no work to charges. B-field do work to magnetic dipoles as we see two magnets attract and collide.

it equals to
$$-\int_A\frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A}$$
using B.
Yes, I'm trying to highlight two points of confusion. You're right I was not clear enough. I hope this helps.

1) I am asking why B-fields do no work on a point charge when the path integral of a charge on a B-field is non-zero? Did we define work as path integral only over conservative fields? Is this why we say B-fields do no work?

2) I am not sure why the closed line integral of B is necessarily zero (if I still had my copy of Boas' Mathematics in the physical sciences, I might know why/if this holds). Additionally, if it is not necessarily zero, I am not sure why the resulting the emf is defined as closed E.dl.

My confusion with Faraday's law in the original question derives from not understanding emf I suspect. Thus, I think ironing out point of confusion 2 will help

A familiar reply to 1) is to see the formula of Lorentz force the part originated from B of which is
$$q \mathbf{v} \times \mathbf{B}$$
and is always perpendicular to velocity of charge q point, ##\mathbf{v}##, thus no work done on the charge.

vanhees71 and ChanceLiterature
Thanks. I realized 1 was very dumb after remembering what conservative and close integral meant.

Additionally, even I'm not sure what I'm asking or saying for 2

Thanks for the help

anuttarasammyak
ChanceLiterature said:
2) I am not sure why the closed line integral of B is necessarily zero
Zero we see as for B is
$$\nabla \cdot \mathbf{B} =0$$
and thus
$$\int_S \mathbf{B} \cdot d\mathbf{S}=0$$
for a closed surface S. Not line integral.

## 1. What is a B-field?

A B-field, also known as a magnetic field, is a region in space where a magnetic force can be detected. It is created by moving electric charges and can exert a force on other moving charges.

## 2. How is Faraday's Law derived from B-fields?

Faraday's Law states that a changing magnetic field can induce an electric field. This can be derived from B-fields using the principle of electromagnetic induction, which states that a changing magnetic field can create an electric field.

## 3. Why is no work done by a B-field?

A B-field does not do work because it is a conservative force. This means that its energy is conserved and does not change as an object moves through it. Therefore, no work is done by the B-field on the object.

## 4. How are B-fields measured?

B-fields can be measured using a device called a magnetometer. This device can detect the strength and direction of a magnetic field and can be used to map out the shape and strength of a B-field in a given region.

## 5. What are some real-world applications of B-fields?

B-fields have many practical applications, such as in electric motors, generators, and MRI machines. They are also used in compasses for navigation and in particle accelerators for scientific research. B-fields are also important in understanding the behavior of the Earth's magnetic field and its impact on our planet.

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