then the
induced emf or the voltage generated by a conductor or coil moving in a magnetic field is
The negative sign describes the direction in which the induced emf drives current around a circuit. However, that direction is most easily determined with a rule known as Lenz’s law, which we will discuss shortly.
The magnetic flux is the amount of magnetic field lines cutting through a surface area
A defined by the unit area vector
If the angle between the unit area
and magnetic field vector
are parallel or antiparallel, as shown in the diagram, the magnetic flux is the highest possible value given the values of area and magnetic field.
Part (a) of
[link] depicts a circuit and an arbitrary surface
S that it bounds. Notice that
S is an
open surface . It can be shown that
any open surface bounded by the circuit in question can be used to evaluate
For example,
is the same for the various surfaces
of part (b) of the figure.
(a) A circuit bounding an arbitrary open surface
S . The planar area bounded by the circuit is not part of
S . (b) Three arbitrary open surfaces bounded by the same circuit. The value of
is the same for all these surfaces.
The SI unit for magnetic flux is the
weber (Wb),
Occasionally, the magnetic field unit is expressed as webers per square meter (
) instead of teslas, based on this definition. In many practical applications, the circuit of interest consists of a number
N of tightly wound turns (see
[link] ). Each turn experiences the same magnetic flux. Therefore, the net magnetic flux through the circuits is
N times the flux through one turn, and Faraday’s law is written as
A square coil in a changing magnetic field
The square coil of
[link] has sides
long and is tightly wound with
turns of wire. The resistance of the coil is
The coil is placed in a spatially uniform magnetic field that is directed perpendicular to the face of the coil and whose magnitude is decreasing at a rate
(a) What is the magnitude of the emf induced in the coil? (b) What is the magnitude of the current circulating through the coil?
A square coil with
N turns of wire with uniform magnetic field
directed in the downward direction, perpendicular to the coil.
Strategy
The area vector, or
direction, is perpendicular to area covering the loop. We will choose this to be pointing downward so that
is parallel to
and that the flux turns into multiplication of magnetic field times area. The area of the loop is not changing in time, so it can be factored out of the time derivative, leaving the magnetic field as the only quantity varying in time. Lastly, we can apply Ohm’s law once we know the induced emf to find the current in the loop.
Solution
The flux through one turn is
so we can calculate the magnitude of the emf from Faraday’s law. The sign of the emf will be discussed in the next section, on Lenz’s law:
The magnitude of the current induced in the coil is
Significance
If the area of the loop were changing in time, we would not be able to pull it out of the time derivative. Since the loop is a closed path, the result of this current would be a small amount of heating of the wires until the magnetic field stops changing. This may increase the area of the loop slightly as the wires are heated.
