Faraday’s and Lenz’s law of Electromagnetic Induction gives the relation between the electric field, magnetic field and induced current.
The following animation is a visual aid to help understand how the law applies:
As you can see in the animation, a conductor is situated in a magnetic field such that it is half-way into it.
If you assume that there are no conductors around then we would expect no current to be induced with this arrangement as we have only two poles on opposite sides of our conductor.
A conducting loop is halfway into a magnetic field However, if a loop of wire is placed in this magnetic field there will be an induced current which seems to travel from the outside into the loop.
Faraday’s and Lenz’s law tells us why this happens. The loop of wire has become an antenna with one end being positive (+) and the other negative (-). The magnetic flux through the conductor is equal to that induced by the current flowing out of it.
The total flux passes through any stationary conductor that is not itself conducting. In this case it goes out of the loop causing a current to circulate around it and back into itself.
1.Apply Lenz’s law:
The direction of “change in magnetic flux” through a circuit loop is the direction that the magnetic field lines are changing at that circuit loop.
The magnetic field lines are oriented according to the right-hand rule. In this case, if we were to view a cross section of our conductor (from above), we would see that there is “no change in current” because there is no change in the number of field lines passing through our loop as it enters and exits.
The induced EMF necessary to maintain constant current (induced by the varying B-field) will be zero and thus no voltage or current will be induced.
2.Apply Faraday’s law:
The equation for magnetic flux through a loop is:
In the case of a “closed circuit” with no current flowing, the magnetic flux would be zero as there are no field lines through/out of the loop.
Therefore, we can say that total flux is equal to zero and thus induced EMF will be equal to zero. In this sense, it is impossible to have an induced EMF in a circuit with no voltage or current flowing.
As you can see below (first animation) there is no induced EMF in a “closed circuit” with nothing changing.
However, based on Faraday’s law (second animation), the induced EMF will equal that of the local B-field.
The term “local” refers to the field at the position of our conductor loop. In this case, since the magnetic flux through our loop is different from that in a region outside of it (it goes in and out through our loop), it is more “local” than total flux (which represents the whole circuit B-field). It also follows that if B-field is changing then so will EMF.
There are a few things to note about this second law: If magnetic flux changes then there will be an induced EMF change which follows from the equation for magnetic flux across a circuit loop.
3.Apply Faraday’s law in reverse:
The equation for Magnetic flux through a loop is equal to “local” EMF. This means that if we know the strength of the B-field at a loop then we can determine what the “local” EMF is.
4.Use the vectors from vector calculus:
Let’s use vector calculus (vectors and their cross products) to rewrite our two laws above. Using the right hand rule, let’s draw out an imaginary line from the magnetic north pole (N) to our conductor
(i). The direction of that line is N and is always perpendicular to the B-field lines. Vectors are added by the “dot product” so that from this line we have:
Note that these two lines are parallel to each other and they both (in vector form) point towards N. The last part of the vector is the length of our imaginary line. It is equal to the area swept out between these two lines. Area = Length * Width and since we don’t know anything about either width or height of our conductor, we can say that the area swept out is equal to Length.
5.Use the right hand rule to determine the total flux:
Now, let’s use the right hand rule to determine our total flux and then relate that to induced EMF. Total flux is . Now we can say that if we know where our loop is located (top left) and the strength of the B-field there then we can find the induced EMF.
6.Use Lenz’s law to determine induced EMF:
Using Lenz law, since we know that current does not change when there is no field change (since no field lines enter or leave the loop) then “total” induction is equal to zero. Therefore, if we know where the loop is located (top left) and the strength of the B-field at that loop then we can easily determine an induced EMF.
