EMF about a loop moving toward a wire with current

• Afterthought
In summary, the conversation discusses finding the emf about a loop without assuming that the magnetic field changes. The problem is not clear and the book gives an answer of bvμI/(2π) * [1/r - 1/(r+a)]. The attempt at a solution includes a calculation for the emf at t = 0 with the wrong sign, but it is determined that the signs do not have a specific meaning. The concept of "B doesn't change" refers to the current not changing and the magnetic field remaining constant at a fixed location in space.

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.

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.

1. What is EMF?

EMF stands for electromagnetic force. It is a force that is generated when an electric current flows through a conductor, such as a wire. This force can cause electrons to move and create an electric field.

2. How does a loop moving towards a wire with current create EMF?

When a loop, or any conductor, moves towards a wire with current, it cuts through the magnetic field created by that current. This cutting of magnetic field lines creates a change in magnetic flux, which induces an EMF in the loop according to Faraday's law of electromagnetic induction.

3. What is the direction of the induced EMF in this scenario?

The direction of the induced EMF follows the right-hand rule. If you point your right thumb in the direction of the conductor's motion, and your fingers in the direction of the magnetic field, then your palm will be facing the direction of the induced EMF.

4. How can the magnitude of the induced EMF be calculated?

The magnitude of the induced EMF can be calculated using Faraday's law, which states that the induced EMF is equal to the rate of change of magnetic flux through the loop. This can be represented by the equation E = -NΔΦ/Δt, where N is the number of turns in the loop, ΔΦ is the change in magnetic flux, and Δt is the time interval over which the change occurs.

5. What factors can affect the strength of the induced EMF in this situation?

The strength of the induced EMF can be affected by several factors, including the speed of the conductor, the strength of the magnetic field, the orientation of the conductor's motion relative to the magnetic field, and the number of turns in the loop. Additionally, the resistance of the conductor and the presence of any external resistors can also impact the strength of the induced EMF.