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

[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.

 

[berkeman]

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?

 

[snatchingthepi]

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.

 

[berkeman]

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?

 

[Delta2]

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 $\vec{F_p}\cdot \vec{dl}=F_pdl\cos(\vec{F_p},\vec{dl})=F_pdl\cos(\frac{\pi}{2}-\theta)=F_pdl\sin\theta$

 

[snatchingthepi]

Thank you all!