# EMF about a loop moving toward a wire with current

## Homework Statement

A picture of a related problem is attached below. My problem is the same, except that v is toward the wire, not away from it. I, r, v, b and a are given. "Find the emf about the loop without assuming that B changes at any point"

My problem is that this question isn't clear to me. Changes at any point of time, or point in space? And if a point in space, do they mean a point on my xy axis, or point inside the loop? The book gives the answer as
bvμI/(2π) * [1/r - 1/(r+a)], which I'm trying to figure out how to get.

## Homework Equations

emf = - d/dt ∫Bda

## The Attempt at a Solution

Here's how I would do the problem normally.
emf = - d/dt ∫ [μI / (2πx)] * [bdx], with limits of integration from r - vt to r - vt + a
= -bμI / (2π) * d/dt [ln(r-vt+a) - ln(r-vt)]
= -bvμI / (2π) * [1/(r - vt) - 1/(r - vt + a)]

This is the same as what the book got, but with t = 0, and the wrong sign . Can someone explain if this is correct, why t = 0, and how it follows from the question? I still don't get the "B doesn't change" condition".

Thanks.

## Answers and Replies

TSny
Homework Helper
Gold Member
I think they want an expression for the emf at the instant when the left side of the loop is a distance r from the current. What is the value of t for this instant?

I think your signs are OK. The overall sign of the emf doesn't have meaning unless you state your convention for "positive" emf. Should the overall sign of the emf change if the loop moves away from the current instead of toward the current?

"B doesn't change" is assuming that the current isn't changing. So, at a fixed location is space, B doesn't change with time. Of course, B does change inside the loop because the loop is changing its location.