# 6.1 Faraday’s Law

The close connection between electricity and magnetism was established by Oersted, who demonstrated that a wire carrying an electric current exerts a force on a compass needle and that the needle always turns to point in the ϕ^ direction when the current is along the z^ direction. The force acting on the compass needle is due to the magnetic field produced by the current in the wire. Following this discovery, Faraday hypothesized that if a current produces a magnetic field, then the converse should also be true: A magnetic field should produce a current in a wire. To test his hypothesis, he conducted numerous experiments in his laboratory in London over a period of about 10 years—all aimed at making magnetic fields induce currents in wires. Similar work was being carried out by Henry in Albany, New York. Wires were placed next to permanent magnets or current-carrying loops of all different sizes, but no currents were ever detected. Eventually, these experiments led to the following discovery by both Faraday and Henry.

To elucidate the induction process, consider the arrangement shown in Fig. 6-1. A conducting loop connected to a galvanometer (a sensitive instrument used in the 1800s to detect current flow) is placed next to a conducting coil connected to a battery. The current in the coil produces a magnetic field B whose lines pass through the loop. In Section 5-4, we defined the magnetic flux Φ passing through a loop as the integral of the normal component of the magnetic flux density over the surface area of the loop, S, or

# Figure 6-1 The galvanometer (predecessor of the ammeter) shows a deflection whenever the magnetic flux passing through the square loop changes with time.

Figure 6 1 Full Alternative Text

Under stationary conditions, the dc current in the coil produces a constant magnetic field B, which in turn produces a constant flux through the loop. When the flux is constant, no current is detected by the galvanometer. However, when the battery is disconnected, thereby interrupting the flow of current in the coil, the magnetic field drops to zero, and the consequent change in magnetic flux causes a momentary deflection of the galvanometer needle. When the battery is reconnected, the galvanometer again exhibits a momentary deflection, but in the opposite direction. Thus, current is induced in the loop when the magnetic flux changes, and the direction of the current depends on whether the flux increases (when the battery is being connected) or decreases (when the battery is being disconnected). It was further discovered that current can also flow in the loop while the battery is connected to the coil if the loop turns or moves either closer to or away from the coil. The physical movement of the loop changes the amount of flux linking its surface S, even though the field B due to the coil has not changed.

A galvanometer is a predecessor of the voltmeter and ammeter. When a galvanometer detects the flow of current through the coil, it means that a voltage has been induced across the galvanometer terminals. This voltage is called the electromotive force (emf), V e m f, and the process is called electromagnetic induction. The emf induced in a closed conducting loop of N turns is given by

V emf=−N d Φ d t=−N d d t∫S B⋅d s (V).(6.6)

Even though the results leading to Eq. (6.6) were also discovered independently by Henry, Eq. (6.6) is attributed to Faraday and known as Faraday’s law. The significance of the negative sign in Eq. (6.6) is explained in the next section.

We note that the derivative in Eq. (6.6) is a total time derivative that operates on the magnetic field B, as well as the differential surface area d s. Accordingly, an emf can be generated in a closed conducting loop under any of the following three conditions:

1. A time-varying magnetic field linking a stationary loop; the induced emf is then called the transformer emf, V e m f t r.

2. A moving loop with a time-varying area (relative to the normal component of B) in a static field B; the induced emf is then called the motional emf, V e m f m.

3. A moving loop in a time-varying field B.

The total emf is given by

V emf=V emf tr+V emf m(6.7)

with V e m f m=0 if the loop is stationary [case (1)] and V e m f t r=0 if B is static [case (2)]. For case (3), both terms are important. Each of the three cases is examined separately in the following sections.