Faraday's Law and Lenz's Law

Faraday's Law

Knowing that a changing magnetic field induced an emf, Faraday went further to analyze what factors affected the magnitude of the induced emf. He found that the following factors influenced the strength of the emf:

Rate of change of the magnetic field strength
1. Example: Inserting a magnet slowly induces a lower emf than inserting the same magnet quickly
Area of the wire loop
1. Example: Decreasing the area of the wire loop decreases the measured induced emf for any given change in magnetic field.
Angle of the loop relative to the magnetic field
1. Example: Maximum emf was induced with the plane of the loop was perpendicular to the direction of the changing magnetic field. Turning the loop such that the angle between the magnetic field and the normal of the loop (the line perpendicular to the surface area of the loop) increased caused a corresponding decrease in induced emf. Once the normal of the loop was perpendicular to the magnetic field, no emf was induced with a changing magnetic field.

EMI _FaradaysLaw_Image.png These three variables describe something known as magnetic flux. For a uniform magnetic field B, the magnetic flux EMI _FaradaysLaw_Equation.gif is defined as:

EMI _FaradaysLaw_Equation1.gif

where B is the strength of the magnetic field, A is the area of the loop, and θ is the angle between the magnetic field and the normal of the loop. Magnetic flux is measured in SI units of webers (Wb), where 1 Wb = 1 T•m2.

To summarize Faradays findings, a change in the magnetic flux through a loop induced an emf. Representing this mathematically gives us Faraday's law of induction:

EMI _FaradaysLaw_Equation2.gif

From a practical stance this shows that by changing any of the three variables related to the magnetic flux (B, A, or θ) an emf will be induced in the coil of wire. The negative in the equation is to show that the direction of the emf is in the opposite direction of the change in magnetic flux. This will be explained more in a moment.

If the coil of wire has multiple turns, the induced emf from each turn will add so we get the modified form of the equation: EMI _FaradaysLaw_Equation3.gif where N is the number of turns in the coil.

Lenz's Law

Where Faraday's law dictates the strength of the induced emf, Lenz's law describes the direction of the induced current due to the induced emf.

Lenz's law states that the current produced by the induced emf moves in a direction, so that its magnetic field opposes the original change in magnetic flux.

EMI _LenzsLaw_Image.png What Lenz's law is telling us is that there is something akin to a magnetic inertia occurring inside the loop of wire when the flux changes. Consider a loop of wire, not connected to a battery or other emf source, that has a bar magnet inside of it. There is a certain magnetic flux through the loop when the magnet is stationary. If the magnet is removed quickly the flux suddenly decreases. In order to compensate, a current is induced in the loop such that the magnetic field it produces is of a strength and pointing in a direction to compensate for the decrease in the original flux. Recall the right hand rule for current carrying loops. If the current flows counter-clockwise, the magnetic field points up through the loop. A clockwise current creates a field pointing in the opposite direction. Combining this understanding with Lenz's law should help you understand why you get a current flowing in one direction in the loop when a magnet is moved into the loop and you get a current flowing in the opposite direction when you remove the magnet. In the above image, you see the north pole of a magnet being inserted into a solenoid. Initially, the solenoid has zero magnetic flux through its center. When a B field pointing to the right is introduced, an opposing B field will be created in the solenoid to attempt to keep the flux at zero. To accomplish this the current must flow through the loop as shown.

Faraday's Law and Lenz's Law Practice

Motional EMF Practice

In the practice above we saw how to calculate an induced emf when the strength of the magnetic field changes through a loop. Faraday's law tells us that an emf can also be induced by simply changing the area of the loop. This video shows an example of this application and uses it to introduce the idea of motional emf.

As you can see in the video a device, known as a commutator, allows for the current in the loop to be reversed every time the loop spins 180 degrees. The electric motor is one of the two most prevalent applications of the relationship between electricity and magnetism. The second device, the electric generator, will be discussed in the next module.

Induction Self-Assessment

A 0.50 T magnetic field is directed perpendicular to the plane of a circular loop of radius 0.25 m. What is the magnitude of the magnetic flux through the loop?
The Earth's magnetic field passes through a square tabletop with a magnitude of 4.95 x 10^-5 T and directed at an angle of 165°relative to the normal of the tabletop. If the tabletop has 1.50 m sides, what is the magnitude of the magnetic flux through it?
A conducting loop has an area of 0.065m² and is positioned such that a uniform magnetic field is perpendicular to the plane of the loop. When the magnitude of the magnetic field decreases to 0.30 T in 0.087 s, the average induced emf in the loop is 1.2 V. What is the initial value of the magnetic field?
A magnetic field is directed perpendicular to the plane of the 0.15 m x 0.30 m rectangular coil comprised of 120 loops of wire. To induce an average emf of -1.2 V in the coil, the magnetic field is increased from 0.1 T to 1.5 T during a time interval Δt. Determine Δt.
A 150 turn solenoid carries a current of 12 A. The radius of the solenoid is 0.050 m and its length is 0.18 m. Determine the magnetic flux through the circular cross-sectional area at the center of the solenoid.
A flat coil with radius 8.0 mm has 50 loops of wire. It is placed in a magnetic field B = 0.30 T in such a way that the maximum flux goes through it. Later, it is rotated in 0.02s to a position such that no flux goes through it. Find the average emf induced between the terminals of the coil.

SOLUTIONS

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