# Where does this term come from? (pulling a wire loop through a B-field)

• snatchingthepi
In summary, the sin(theta) term in the attached picture from Griffiths' 4th edition EM text comes from the dot product between the drift velocity of a charge and the total velocity of the charge. This is because the element of the integral, dl, is in the direction of the total velocity, and the angle between dl and F_p is the angle between the total velocity and F_p. Therefore, the angle between dl and F_p is pi/2 - theta, and by definition of the dot product, F_p * dl = F_p * dl * sin(theta).
snatchingthepi
Homework Statement
Pulling a hoop through a uniform B-field
Relevant Equations
emf = loopintegral (f_pull dot dl)
I can't for whatever reason figure out where the sin(theta) term is coming from in the attached picture of page 306 of Griffiths' 4th edition EM text. The paragraph says it comes from the dot product, but I just don't see where it's coming from.

Can you also scan the figure that this is referring to? Theta must be the angle of the loop with respect to the B-field direction?

berkeman said:
Can you also scan the figure that this is referring to? Theta must be the angle of the loop with respect to the B-field direction?

Yes here it is.

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So the dot product enters in because if the coil is oriented parallel to the B-field, then none of the flux pierces the plane of the loop. Does that make sense?

snatchingthepi
yes it all fits now, ##\theta## is the angle between ##\vec{u}## (the drift velocity of a charge inside the conductor) and the total velocity of the charge ##\vec{w}##. The element ##\vec{dl}## of the integral is in the direction of the total velocity ##\vec{w}## (it is ##\vec{dl}=\vec{w}dt##), because as Griffith says to find the work of ##F_p## we have to follow a charge at its journey around the loop, and this journey is done with the total velocity ##\vec{w}##. Thus the angle between ##\vec{dl}## and ##\vec{F_p}## is the angle between ##\vec{w}## and ##\vec{F_p}## which is ##\frac{\pi}{2}-\theta## .Thus and by definition of dot product ##\vec{F_p}\cdot \vec{dl}=F_pdl\cos(\vec{F_p},\vec{dl})=F_pdl\cos(\frac{\pi}{2}-\theta)=F_pdl\sin\theta##

Last edited:
snatchingthepi
Thank you all!

berkeman and Delta2

## 1. Where did the term "pulling a wire loop through a B-field" originate?

The term "pulling a wire loop through a B-field" originated from the study of electromagnetism and the behavior of electric currents in the presence of a magnetic field. It is often used to describe an experimental setup in which a wire or conductor is moved through a magnetic field to induce an electric current.

## 2. What is the significance of the B-field in this term?

The B-field, also known as the magnetic flux density, is a measure of the strength of the magnetic field at a specific point in space. In this term, it refers to the magnetic field that the wire loop is being pulled through. The B-field is essential in inducing an electric current in the wire.

## 3. How does pulling a wire loop through a B-field produce an electric current?

When a wire or conductor is moved through a magnetic field, it cuts through the lines of magnetic flux, which generates a force that pushes electrons in the wire. This movement of electrons creates an electric current in the wire.

## 4. What is the practical application of pulling a wire loop through a B-field?

This experimental setup has practical applications in various fields, such as generating electricity in power plants and motors, as well as in induction cooking and magnetic levitation technology.

## 5. Are there any limitations to pulling a wire loop through a B-field to induce an electric current?

Yes, there are limitations to this method as the strength of the magnetic field, the speed of the movement, and the dimensions of the wire can all affect the amount of current induced. Additionally, the wire must be a continuous loop to allow for the flow of electrons.

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