# Problem on induced EMF

Tags:
1. Jan 24, 2015

### arpon

1. The problem statement, all variables and given/known data

Suppose, there is a 'constant' magnetic field $B$ in the upward direction. A loop of conductive wire(XYZ in the picture, which is connected to a closed circuit with resistance R) is placed horizontally on it. The area of the loop is being increased with time. Will there be any current flow in the circuit?

2. Relevant equations
$\mathcal {E} = -\frac{d \phi}{dt}$
$\phi = BA cos \theta$

3. The attempt at a solution
I think, as the area is changing, there will be a current flow proportional to the rate of change of area inclosed by the loop.

2. Jan 24, 2015

### BvU

It's good to think of this induced emf as a consequence of the Lorentz force on the charge carriers in the wire. For the loop the velocity of the wire segments is outwards, perpendicular to B. The Lorentz force is $\propto \vec v \times \vec B$, so clockwise.

The xyz directions require some explanation...
And $\Delta \Phi$ due to moving the outer loop into the B field (and changing its area as well ?) is a complication I disregarded.

3. Jan 24, 2015

### arpon

Can't I calculate the induced EMF in this way? :
$\mathcal E = \frac{d\Phi}{dt} = \frac{d(BA)}{dt} = B \frac{dA}{dt}$
And how can I calculate the induced EMF or current flow by the formula you mentioned: $\vec F = q \vec v \times \vec B$ [$q$ is the total charge of the charge carriers in the wire]
What kind of explanation?

4. Jan 24, 2015

### BvU

Yes you can. dA is for the entire loop, not just the circle part.

Hard work. It's done for you here . Wire is neutral, of course. But there are mobile charge carriers and immobile ones. You refer to the former, of course.

A little better than just writing x, y and z around the circular loop. You do indicate that B is in the z direction. Add that the circular loop is in the xy plane. (And you can avoid the complication I mentioned by placing the rectangular part of the loop in the yz plane, so that that part of $\Phi=0$

5. Jan 24, 2015

### arpon

Actuallly, I named the points on the circular loop as x,y,z ; I realize it was not wise, because it creates confusion with the x,y,z-axes. However, I am explaining it again, the magnetic field is in the Z direction. The rectangular part of the circuit is on the yz-plane, and the circular part(which is changing the area enclosed by itself) is on the xy-plane, i.e., perpendicular to the direction of magnetic field.
Now, I would request you to revise your posts keeping in mind this explanation.

6. Jan 24, 2015

### BvU

As far as I can distinguish the way you intended the configuration is indeed the way I understood it. So: yes, the movement of the wire increasing the diameter of the loop induces an electromotive force. And it can be calculated using Faraday's law $$\mathcal E = \oint_{\partial \Sigma} \vec E \cdot d\vec \ell= -{d\over dt}\;\int_\Sigma \vec B \cdot \vec{dA}$$which comes down to $B\; \frac{dA}{dt}$