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Motional EMF

When the conducting bar in the figure moves to the right through the magnetic field B conduction electrons experience a magnetic force directed downward. Hence some of the conduction electrons accumulate at the bottom of the bar, leaving an excess positive charge at the top. See figure (a) at the right. The accumulation of electrons at the bottom of the bar continues until the electrostatic force of attraction between the negative and positive charges at the ends of the bar balances the magnetic force on the conduction electrons. If a resistance R is connected across the ends of the conducting bar a current I will flow through that resistance. Hence the moving bar acts as a seat of electromotive force (emf).

The separated charges on the ends of the moving conductor give rise to an induced emf, called a motional emf, because it originates from the motion of charges through a magnetic field. The emf exists as long as the rod moves. If the rod is brought to a halt, the magnetic force vanishes, with the result that the attractive electric force reunites the positive and negative charges and the emf disappears. The emf of the moving rod is analogous to that between the terminals of a battery. However, the emf of a battery is produced by chemical reactions, whereas the motional emf is created by the agent that moves the rod through the magnetic field (like the hand in figure (b)).

Recall that emf is defined as the work per unit charge done by the seat of emf when it moves the conventional positive charge internally from the negative terminal to the positive terminal. In a chemical battery this work comes from the chemical energy stored in the battery; in a solar cell this work comes from sunlight. In the example above this work is done by the external force that moves the bar through the magnetic field. Since the bar carries an upward current I through the magnetic field B it experiences magnetic force directed to the left of magnitude where here  = 90: . Assume that the hand exerts a force that just balances the magnetic force, so that the bar moves with constant velocity. (We assume that the conducting rails are frictionless.) Let the bar move a distance x to the right in time t. The work done by the external force is

.

But , the charge that moves through the conducting bar in time t. Also, , the speed of the moving bar. Hence

If the velocity v of the bar makes and angle  with the magnetic field B then . Note that the distance L is always measured perpendicular to the magnetic field lines.E is called the induced (or motional) emf. It is measured in volts. (Verify the units.)

Consider a side view from above of the current loop with the conducting rod shown in the previous figure. In figure (c) the magnetic field B makes a right angle with the velocity v of the conducting bar. In figure (d) the magnetic field makes an angle  with the velocity v of the conducting bar, or equivalently an angle  with a line that is normal to the current loop. In terms of the angle  the induced emf can be written as

Magnetic Flux

It is convenient to express the induced emf in terms of magnetic flux. Consider a magnetic field passing through an area A such that the field makes an angle  with respect to a line normal to A. See figure (e). The magnetic flux through this surface area is defined as

The SI units of flux are Tm2. 1 Tm2 = 1 weber (Wb).

We can now write the induced emf of the previous example as follows. Assume the magnetic field makes an angle  with respect to a line normal to the current loop as in figure (d). Let the conducting bar move a distance x to the right in time t. In this time the conducting bar sweeps out an area A. See figure (f). Since the induced emf can be written as

But , the change in magnetic flux through the current loop. Thus we can write

where a minus sign has been introduced to remind us that the direction of the induced current (or the polarity of the induced voltage) is such as to oppose the change that produced it. This formula is Faraday’s Law. Faraday’s law turns out to be generally true, regardless how the magnetic flux changes. And note that the magnetic flux can change in three different ways: (i) through a change in the magnitude of the magnetic field B, (ii) through a change in the area A through which the magnetic field passes and (iii) through a change in the angle  the magnetic field makes with respect to a normal line through A. In this course we will see examples of an emf induced in each of these three different ways.