PhysicsLAB: Induced EMF

Magnetic Flux

The number of magnetic field lines, or magnetic flux, which pass through a given cross-sectional area can be calculated with the formula

Φ = B_{perpendicular}A

where

Φ is the number of flux lines, measured in webers (Wb)
B_{perpendicular} is the magnetic field strength, measured in tesla (T)
A is the cross-sectional area, measured in m²

This relationship defines a tesla to be a weber/m².

As shown below by two extreme cases, the number of flux lines can vary from a maximum value of Φ = BA when the area vector, A, is perpendicular to the magnetic field lines, B, to a minimum value of zero when the area vector, A, is parallel to the field lines, B.


Φ = BA (maximum value)	Φ = B(0) = 0 (minimum value)

Earlier we saw that moving electric charges in electric currents can create magnetic fields. In the early 1830's Faraday used the concept of flux lines to explain his experimental results that an emf can be induced in a coil when it is exposed to a changing magnetic flux.

Refer to the following information for the next question.

A magnetic field having a magnitude of 4.0 x 10^-2 T passes diagonally through a circular coil of wire that has a radius of 10 cm. The angle between the magnetic field lines and the coil's area vector is 37º.

	How many flux lines will pass through the coil?

Faraday's Law

As stated earlier, changing flux is the foundation of Faraday's Law of Induction.

ε = -N(ΔΦ/Δt)

where

ε is the induced voltage in a coil, measured in volts
N is the number of loops in the coil
ΔΦ is the changing flux, measured in webers
Δt is the time over which the change occurs, measured in seconds

When the number of flux lines is constant, no emf is induced in a coil.

The expression

ΔΦ/Δt = Δ(B_{perpendicular}A)/Δt

is merely an abbreviation for the rate of change of flux lines. Notice, an emf is induced only when the number of flux lines passing through an enclosed area changes. This number can be increases or decreased either

by changing the strength of the magnetic field OR
by changing the area of the coil.

Note that the induced emf opposes these changes as designated by the negative sign in this formula.

Let's examine the following physlet that shows us how the flux through a coil can be changed by moving a bar magnet through a coil. Notice the curve in the graph always remains positive since the physlet is always tracking flux lines directed towards the left. Remember that field lines exit the north pole of a bar magnet and enter the south pole.

Refer to the following information for the next eight questions.

Now, instead of using a magnet as a source of flux lines, let's use a primary coil whose current we can control with a switch. Let's discuss what will happen in the second circuit as we close and then open the switch.

	Before the switch is closed, is there any current flowing in either circuit?

	Instantaneously, when the switch is closed, in which direction will current begin to flow in the top circuit?

	Using the right hand curl rule, what is the direction of the magnetic moment of the top coil?

	How does the bottom coil respond to the incoming flux lines from the top coil?

	In which direction will induced currents begin to flow in the bottom circuit?

	After the switch has been closed for a while, what happens to the current in the second circuit?

	If the switch were to be opened, how would the bottom coil respond?

	What would a graph of current vs time look like in the second circuit?

Now let's examine a second physlet that produces a graph similar to our example's current vs time graph. In this physlet the flux through the coil is being changed by moving the North pole of a bar magnet initially towards a coil and then back away. Notice how the direction of the induced voltage in the coil changes to oppose the motion of the magnet.

A statement of Faraday's Law that is used when the cross-sectional area of the loop remains constant, but the magnetic field strength is changing is

ε = -NA(ΔB/Δt)

where

ε is the induced emf in the coil measured in volts
ΔB/Δt is the rate of change in the magnetic field: (B_f - B_o)/t
B is measured in teslas,
t is in seconds,
N is the number of loops (usually one),
A is the cross sectional area (rectangle: A = lw, or a circle: A = πr²)

Refer to the following information for the next two questions.

A square loop of area A is placed on top of a variable magnet whose field strength versus time is plotted on the graph shown.

	When would the loop experience an induced voltage?

	Would the currents in the loop flow in the same direction during the intervals BC and CD?