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
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.