Magnetic Induction

(197) |

Suppose that the magnetic field changes in time, causing the
magnetic flux linking circuit to vary.
Let the flux change by an amount in the time interval . According to Faraday's law, the emf
induced around loop is given by

Suppose that , so that the emf acts in the positive direction.
How, exactly, is this emf produced? In order to answer this question,
we need to remind ourselves what an emf actually is. When we say that
an emf acts around the loop in the positive direction,
what we really mean is that a charge which circulates once around
the loop in the positive direction acquires the energy .
How does the charge acquire this energy? Clearly, either an electric
field or a magnetic field, or some combination of the two, must perform the
work on the charge as it circulates around the loop.
However, we have already seen, from Sect. 8.4, that a magnetic
field is unable to do work on a charged particle. Thus, the charge must
acquire the energy from an *electric* field as it
circulates once around the loop in the positive direction.

According to Sect. 5, the work that the electric field does on the charge as it goes around
the loop is

(199) |

The term on the right-hand side of the above expression can be recognized as the

Equations (198) and (200) can be combined to give

Equation (201) describes how a time-varying magnetic field *generates*
an electric field which fills space. The strength of the electric field is directly
proportional to the rate of change of the magnetic field. The
field-lines associated with this electric
field form loops in the plane perpendicular to the magnetic field. If the
magnetic field is increasing then the electric field-lines circulate in the
opposite sense to the fingers of a right-hand, when the thumb points
in the direction of the field. If the
magnetic field is decreasing then the electric field-lines circulate in the
same sense as the fingers of a right-hand, when the thumb points
in the direction of the field. This is illustrated in Fig. 35.

We can now appreciate that when a conducting circuit is placed in a
time-varying magnetic field, it is the electric field induced by the changing
magnetic field which gives rise to the emf around the circuit. If the loop has a
finite resistance then this electric field also drives a current around the circuit.
Note, however, that the electric field is generated irrespective of the
presence of a conducting circuit. The electric field generated by a time-varying
magnetic field is quite different in nature to that generated by a set
of stationary
electric charges. In the latter case, the electric field-lines begin on
positive charges, end on negative charges, and *never* form closed loops
in free space. In the former case, the electric field-lines *never* begin or
end, and *always* form closed loops in free space. In fact, the electric
field-lines generated by magnetic induction behave in much the same
manner as magnetic field-lines. Recall, from Sect. 5.1, that an electric
field generated by fixed charges is unable to do net work on a charge
which circulates in a closed loop. On the other hand, an electric
field generated by magnetic induction certainly can do work on a charge
which circulates in a closed loop. This is basically how a current is induced in
a conducting loop placed in a time-varying magnetic field. One consequence of this
fact
is that the work done in slowly moving a charge
between two points in an inductive electric field *does* depend on the
path taken between the two points. It follows that we cannot
calculate a *unique* potential difference between two points in an inductive
electric field. In fact, the whole idea of electric potential breaks down
in a such a field (fortunately, there is a way of
salvaging the idea of electric potential in an inductive field, but this topic
lies beyond the scope of this course). Note, however, that it is still
possible to calculate a *unique* value for the emf generated around a conducting
circuit by an inductive electric field, because, in this case, the path taken by
electric charges is uniquely specified: *i.e.*, the charges have
to follow the
circuit.