Induced EMF in a moving Loop Conductor

[Kharrid]

Homework Statement

A circular wire loop is placed in a uniform magnetic field. Find if there is an induced current. The normal of the loop points along the positive x-axis and the magnetic field also points along the positive x-axis.

1. loop moves to the right

2. loop moves to the left

Relevant Equations

F = qv x B
Flux = BAN
emf = ∫ E ds = -d Φ / dt

I am having trouble figuring out if the circular loop has an induced current.

One explanation is ∫ E ds = -d Φ / dt. Since flux = B ⋅ A, a change in the magnetic field would require a change in the magnetic field, a change in the area, or change in direction of either vector. Since none of these happen, the flux while the loop is moving to the right is constant and no electric field is induced that would create an induced current.

After looking online, it seems that there is always an induced emf when a loop is moved through a uniform magnetic field but I'm not sure why. If I try to follow the logic from my book, I'm supposed to find the induced electric field that causes the charges to move. Well, the induced electric field is the negative rate of change in flux. Since the magnetic field is uniform, the area vector is the same, and there is no change in direction between the vectors while the loop is moving, there is no change in flux. Hence, no induced emf.

Am I missing something?

The exact "online link" is https://www.physicsforums.com/threads/induced-emf-when-accelerates-in-uniform-magnetic-field.515489/

 

[etotheipi]

I agree with your analysis using Faraday's law. The flux linked is constant.

The link you attach is referring to a rod moving through a magnetic field. The emf induced in a rod if the rod, magnetic field and velocity vector are all mutually orthogonal is $\mathcal{E} = Blv$.

You might consider your loop to be a rectangular frame made up of 4 individual wires/rods, with one pair of sides parallel to the velocity vector and the other two orthogonal to the velocity vector. Suppose the magnetic field is normal to the plane of the loop. What is the EMF induced in each loop (pay attention to their directions...)? What is the total EMF around the loop?

 

[Kharrid]

The work is in the photo. Essentially, I think the emf of the two horizontal sides is 0 and the emf of the two vertical sides is vBL. Since the vertical sides have the same direction and magnitude, they cancel and the total emf is zero around the loop. Is this correct?

 

[etotheipi]

That's essentially correct, yes. Hopefully that satisfies you that everything is still consistent!

The way I like to think of it is in terms of the vector triple product, $\mathcal{E} = (\vec{v} \times \vec{B}) \cdot \vec{l} = vBl\sin{\theta}\cos{\phi}$. If all 3 vectors are mutually orthogonal this reduces to $\mathcal{E} = Blv$. If at least two of $\vec{v}$, $\vec{l}$, and $\vec{B}$ are parallel, the whole expression becomes zero.