Magnetic Flux and Lenz's Law


Both magnetic flux and Lenz's Law are fundamental aspects of electromagnetism, and will appear on the test. Previous concepts pertaining to magnetism should already be known before attempting to learn this. Magnetic flux is proportional to the component of the magnetic field vector perpendicular to the area. A change in magnetic flux induces current in a wire, which must oppose the change in flux, according to Lenz's Law. These concepts are important in order to understand the relationship between electricity and magnetism.

Magnetic Flux



Magnetic flux can be described as how much of the field flows through a specific area. It is represented by:
ϕB=B⋅A=BAcosθ
where B represents the magnetic field and A represents the area of the surface. The unit for magnetic flux is the Weber (Volt-seconds, or Tesla-meters squared). For basic purposes, magnetic flux can be viewed as proportional to the number of field lines passing through the surface. However, in reality it is the component of the field vector through a surface. The flux is at its maximum when the vector for magnetic field and the area face the same direction (that is, when the angle between the two is zero), and is zero when the magnetic field vector is perpendicular to the area.



28_11_Magnetic_Flux.jpg
The magnetic flux is proportional to the component perpendicular to the area.


Example 1

A magnetic field of 0.2 Tesla passes through a loop with a radius of 3cm. What is the magnetic flux when the loop is at a).0 degrees b).45 degrees c).90 degrees?


Lenz's Law


Faraday's Law of Electromagnetic Induction

Any changes in magnetic flux induce a change in emf. Faraday's Law of Electromagnetic Induction states that the induced emf in any closed circuit is equal to the time rate of change of the magnetic flux through the circuit. This is equal to:
|ε|=N|(ΔϕB)/(Δt)|
where ε=EMF, ΔϕB= the change in flux, and Δt= the change in time, and N is the number of turns in the coil.

fday.gif
Illustration of Faraday's Law

Lenz's Law

Lenz's Law is a form of Conservation of Energy that states that the induced current will always flow in the direction so that its magnetic field opposes the change in magnetic flux. This is indicated by adding minus sign to Faraday's Law.
ε=-N(ΔϕB)/(Δt)
If the direction did not oppose the change in flux, then the Conservation of Energy would be violated because without the reversed polarity, the EMF would add to the total energy which is coming from nowhere, a clear violation of this principle. For example, if the current in the loop mentioned in the previous example had gone the same direction as the change in magnetic flux, the magnetic force would increase, causing a net gain of potential energy that is coming from absolutely nowhere.

Note that if magnetic flux does not change, then the induced emf will be zero, which means that no current can flow through the circuit. Thus, when the circuit and the magnetic field are stationary, there will be no induced current.


lenz.gif
Visualization of Lenz's Law


Example 2

The loop from the previous example is rotated 45 degrees in .5 seconds. What is the magnitude of the induced emf in the coil?



Example 3

A uniform bar magnet is dropped into a coil of wire such that the north pole is facing downward. Describe the behavior of the induced current in the coil of wire during this time.

Answers


Example 1

Use the equation ϕ=BAcosθ to find the flux in each through each loop.
a.) ϕ=(0.2T)(π)(0.03)2 cos(0)=5.65*10-4 Wb
b.) ϕ=(0.2T)(π)(0.03)2 cos(45)= 4.00*10-4 Wb
c.) ϕ=(0.2T)(π)(0.03)2 cos(90)= 0 Wb

Example 2

The change in flux can be found from the first example, and just use the equation |ε|=|(Δϕ)/(Δt)|.
ε=|(4.00*10-4 Wb-5.65*10-4 Wb)/(0.1s)|=1.65*10-3 V

Example 3

When the magnet is dropped into the coil, the downward-facing north pole has field lines being emitted from it. Thus, as the center falls closer to the coil, the field lines increase, which is a net increase in the field going downward. Because of this, the flux increases in the downward direction. Because of Lenz's Law, the induced current must oppose this change in flux, so the induced field must point upward. Using the magnetic hand rule for coils, the current is found to flow counter-clockwise in order for this field to exist.
magnetloop1.gif
As the magnet enters the coil, the downward flux increases, making the induced current flow counter-clockwise in order for Lenz's Law to be upheld.

When the center of the magnet passes through the coil, the field is no longer increasing in the downward direction becase the coil is now getting closer to the south pole than the north pole. This mean that the field increases in the upward direction, which means the flux has to increase in the upward direction. Once again, using Lenz's Law, it is detemined that the induced field must face in the downward direction. Using the magnetic hand rule, it is determined that the current must flow clockwise in order for this field to exist.
magnetloop2.gif
After the center of the magnet passes through the coil, the upward flux increases, making the induced current flow clockwise in order for Lenz's Law to be upheld.

Note that if the magnet was held stationary to the coil, than the flux would not increase, and there would be no current in the coil. It also doesn't matter whether the magnet or the coil was moved in this way, the effect would be exactly the same.


Sources



Hyperphysics (Magnetic Flux)
Hyperphysics (Lenz's Law)
Wolfram (Magnetic Flux)
Wolfram (Lenz's Law)
Wolfram (Faraday's Law)
Magnetic Flux
Lenz's Law (Good Animations)


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