# Faraday's law -- How is the RHS required for all surfaces?

• millo
In summary, the independence of the right side of ∫c Edl =-d/dt∫sBdA on the surface s, as long as the curve c remains the same, is due to Stoke's Theorem. This is possible because of Gauss's law for magnetism, which allows us to set B equal to the curl of A. Therefore, the surface integral of B over S will always be equal to the line integral of A over the curve C, resulting in a constant value for all surfaces S with the same boundary C.
millo
Homework Statement
My question is how do we know that faraday's law right side is required for all surfaces.
Relevant Equations
∫_c Edl =-d/dt ∫_s Bda
c Edl =-d/dt∫sBda

do you mean why the right side is independent of the surface s as long as we keep the curve c , the boundary of the surface the same? This is a consequence of Stoke's Theorem (or curl theorem).

Since ##\nabla\cdot \vec{B}=0## (Gauss's law for magnetism) we can set ##\vec{B}=\nabla\times\vec{A}##. Stokes theorem tell us that $$\iint_S \vec{B}\cdot d\vec{S}=\iint _S(\nabla\times \vec{A})\cdot d\vec{S}=\oint_C\vec{A}\cdot d\vec{l}$$ so the surface integral of B over S will equal the line integral of A over the curve C, thus it remains constant for all surfaces S with the same boundary C.

millo
Yes this is what I meat, thanks for answer.

berkeman and Delta2

## 1. What is Faraday's law?

Faraday's law is a fundamental principle in electromagnetism that describes the relationship between a changing magnetic field and an induced electric field. It states that the magnitude of the induced electric field is proportional to the rate of change of the magnetic field over time.

## 2. What is the RHS in Faraday's law?

The RHS, or right-hand side, in Faraday's law refers to the mathematical equation that represents the relationship between the changing magnetic field and the induced electric field. It is written as E = -dΦ/dt, where E is the induced electric field, Φ is the magnetic flux, and t is time.

## 3. Why is the RHS required for all surfaces?

The RHS is required for all surfaces because Faraday's law applies to any surface that is enclosed by a closed loop. This means that the induced electric field is present on all surfaces, not just the ones that are in direct contact with the changing magnetic field.

## 4. How is the RHS used in practical applications?

The RHS is used in practical applications to calculate the induced electric field in various situations. This is important for understanding and predicting the behavior of electromagnetic devices, such as generators and motors, as well as for studying phenomena such as electromagnetic induction and electromagnetic waves.

## 5. Are there any exceptions to Faraday's law and the RHS?

There are some exceptions to Faraday's law and the RHS, such as when the magnetic field is constant or when the surface is not completely enclosed by a closed loop. In these cases, the induced electric field may not be present or may be significantly different from what is predicted by the RHS.

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