# Induction in an inflating loop in constant B?

• Lojzek
In summary, when a wire moves in a constant magnetic field, there is no electric field or voltage present. However, charges experience a magnetic force and can perform work on a charge after completing a circle. Two methods, using the magnetic force and Faraday's law of induction, can be used to calculate this work. While the second method may seem incorrect for a constant B and changing loop area, it is in fact valid and can be derived for any change in the flux.
Lojzek
A loop is made from a conductive wire. The wire moves, so the area inside the loop is time dependent: S=S(t)
There is a constant homogeneus magnetic field B directed perpendicular to the wire and we are supposed to calculate induced voltage.

In my opinion there is no electric field and no voltage, since there field B is
constant. However there is a magnetic force experienced by charges moving in magnetic field:

Method 1:

F=e*B*v

If this force is integrated over the loop to gain work on a charge e after 1 circle, we get:

A=-e*B*dS/dt

The proposed solution used Faraday's law of induction:

Method 2:

U=-dfi/dt=-d(B*S)/dt=-B*dS/dt

I think that this is a misuse of the law, since corresponding Maxwell's equation can be
used only for fixed loop, but changing magnetic field. However the work gained by a charge completing one circle is exactly the same as with previous method:

A=e*U=-e*B*dS/dt

My question is:

Is the method 2 really incorrect? If yes, why is the work the same? If no, how do we prove that Faraday's law can be used in case of constant B and changing loop area?

Last edited:
Faraday's works for any change in the flux.
For constant B and changing area, it is often derived for a rectangle in elementary texts.
It is derived for an arbitrary change in the area in more advanced texts.

I found some related texts and I hope I understand the problem now: it seems that
in case of the moving loop the "induced voltage" is not an integral of a real electric field, but an integral of a fictional electric field, that would do the same work on a circling charge.

## 1. What is induction in an inflating loop in constant B?

Induction in an inflating loop in constant B refers to the process of generating an electric current in a loop of wire as it moves through a magnetic field that has a constant strength (B). This phenomenon is also known as electromagnetic induction and is the basis for the operation of generators and motors.

## 2. How does induction occur in an inflating loop in constant B?

Induction occurs in an inflating loop in constant B when the loop moves through the magnetic field, causing a change in the magnetic flux through the loop. This change in flux induces an electric current in the loop, according to Faraday's law of induction.

## 3. What factors affect the magnitude of induced current in an inflating loop in constant B?

The magnitude of the induced current in an inflating loop in constant B depends on several factors, including the strength of the magnetic field, the speed at which the loop moves through the field, and the size and shape of the loop.

## 4. How is the direction of the induced current determined in an inflating loop in constant B?

The direction of the induced current is determined by Lenz's law, which states that the direction of the induced current will be such that it opposes the change in magnetic flux through the loop. This means that the induced current will create a magnetic field that opposes the original magnetic field.

## 5. What are some practical applications of induction in an inflating loop in constant B?

Induction in an inflating loop in constant B has many practical applications, including electricity generation in power plants, operation of electric motors, and wireless charging of electronic devices. It is also used in a variety of scientific and technological fields, such as magnetic levitation and electromagnetic braking.

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