30.5 Faraday’s Law

Charges don’t start moving spontaneously. A current requires an emf to provide the energy. We started our analysis of induced currents with circuits in which a motional emf can be understood in terms of magnetic forces on moving charges. But we’ve also seen that a current can be induced by changing the magnetic field through a stationary circuit, a circuit in which there is no motion. There must be an emf in this circuit, even though the mechanism for this emf is not yet clear.

The emf associated with a changing magnetic flux, regardless of what causes the change, is called an induced emf $ε$ . Then, if there is a complete circuit having resistance R, a current

I_{induced} = \frac{ε}{R}

(30.13)

is established in the wire as a consequence of the induced emf. The direction of the current is given by Lenz’s law. The last piece of information we need is the size of the induced emf $ε$ .

The research of Faraday and others eventually led to the discovery of the basic law of electromagnetic induction, which we now call Faraday’s law. It states:

In other words, the induced emf is the rate of change of the magnetic flux through the loop.

As a corollary to Faraday’s law, an N-turn coil of wire in a changing magnetic field acts like N batteries in series. The induced emf of each turn of the coils add, so the induced emf of the entire coil is

ε_{coil} = N | \frac{d Φ_{per coil}}{d t} | (Faraday ’ s law for an N - turn coil)

(30.15)

As a first example of using Faraday’s law, return to the situation of Figure 30.5, where a wire moves through a magnetic field by sliding on a U-shaped conducting rail. FIGURE 30.26 shows the circuit again. The magnetic field $\vec{B}$ is perpendicular to the plane of the conducting loop, so $θ = 0 °$ and the magnetic flux is $Φ = A B$ , where A is the area of the loop. If the slide wire is distance x from the end, the area is $A = x l$ and the flux at that instant of time is

Φ_{m} = A B = x l B

(30.16)

The figure illustrates a sliding wire placed on a circuit having a magnetic field.

FIGURE 30.26 Full Alternative Text

The flux through the loop increases as the wire moves. According to Faraday’s law, the induced emf is

ε = | \frac{d Φ_{m}}{d t} | = \frac{d}{d t} (x l B) = \frac{d x}{d t} l B = v l B

(30.17)

where the wire’s velocity is $v = d x / d t$ . We can now use Equation 30.13 to find that the induced current is

I = \frac{ε}{R} = \frac{v l B}{R}

(30.18)

The flux is increasing into the loop, so the induced magnetic field opposes this increase by pointing out of the loop. This requires a ccw induced current in the loop. Faraday’s law leads us to the conclusion that the loop will have a ccw induced current $I = v l B / R$ . This is exactly the conclusion we reached in Section 30.2, where we analyzed the situation from the perspective of magnetic forces on moving charge carriers. Faraday’s law confirms what we already knew but, at least in this case, doesn’t seem to offer anything new.