Peoples Physics Concepts

http://www.ck12.org Chapter 14. Magnetism

In the example above, there are four loops of wire(N= 4 )and each has areaπr^2 (horizontally hashed). The

magnetic field is pointing at an angleθto the area vector. If the magnetic field has magnitudeB, the flux through

the loops will equal 4 cosθBπr^2. Think of the magnetic flux as the part of the “bundle” of magnetic field lines

“held” by the loop that points along the area vector.

If the magnetic flux through a loop or loops changes, electrons in the wire will feel a force, and this will generate a

current. Theinduced voltage(also calledelectromotive force, or emf) that they feel is equal to the change in flux

4 Φdivided by the amount of time 4 tthat change took. This relationship is called Faraday’s Law of Induction:

em f=−

∆Φ

∆t

Faraday’s Law of Induction

The direction of the induced current is determined as follows: the current will flow so as to generate a magnetic

field thatopposesthe change in flux. This is called Lenz’s Law. Note that the electromotive force described above

is not actually a force, since it is measured in Volts and acts like an induced potential difference. It was originally

called that since it caused charged particles to move — henceelectromotive— and the name stuck (it’s somewhat

analogous to calling an increase in a particle’s gravitational potential energy difference a gravitomotive force). For

practical (Ohm’s Law, etc) purposes it can be treated like the voltage from a battery.

Since only a changing flux can produce an induced potential difference, one or more of the variables in equation [5]
must be changing if the ammeter in the picture above is to register any current. Specifically, the following can all
induce a current in the loops of wire:

• Changing the direction or magnitude of the magnetic field.


• Changing the loops’ orientation or area.


• Moving the loops out of the region with the magnetic field.

Example 1

You are dragging a circular loop of wire of radius .25 m across a table at a speed of 2 m/s. There is a 2 m long

region of the table where there is a constant magnetic field of magnitude 5 T pointed out of the table. As you drag

the loop across the table, what will be the induced Emf (a) as the loop enters the field (b) while it is in the field and

(c) as it exits the field.

Solution

(a): As the loop enters the field, the flux will start at zero and begin to increase until the loop is entirely inside the

field. The flux will increase from 0 Tm^2 to some maximum value in the time it takes for the loop to move into the

field. We can find this maximum value using the dimensions of the loop and the strength of the magnetic field. The

dot product will be equal to one since the area and magnetic field vectors are parallel.

Φf=NBA

Φf= 1 ∗5 T∗π(.25 m)^2

Φf=.98 Tm^2

Since we also know the radius of the loop and the speed at which it is being pulled, we also can find out how long it
will take for the loop to move within the magnetic field.