48 Motional Emf

Motion as a Source of Induced Voltage

In the previous section, we learned that a changing magnetic flux produces an induced voltage (emf). One of the most common ways to change magnetic flux is simply to move a conductor through a magnetic field. This process is called motional emf, and it is one of the fundamental principles behind electric generators, many medical devices, and numerous technologies used every day.

You have already encountered one example of motional emf when studying the Hall effect. As electric charges move through a magnetic field, they experience the magnetic force

This force separates positive and negative charges, creating a voltage across the conductor. In the Hall effect, this voltage can be used to measure magnetic field strength or the speed of moving charges. Motional emf is a broader application of the same physical principle and provides a practical way of converting mechanical energy into electrical energy.

Motional Emf in a Sliding Conductor

Consider the system shown in Figure 48.1. A conducting rod moves to the right with speed [latex]v[/latex] along two conducting rails separated by a distance [latex]\ell[/latex]. The entire system is immersed in a uniform magnetic field [latex]B[/latex] directed into the page. The rails are connected through a resistor [latex]R[/latex], forming a complete electrical circuit.

As the rod moves, the rectangular area enclosed by the rails, rod, and resistor becomes larger. Since the magnetic field remains constant while the enclosed area increases, the magnetic flux through the circuit also increases. According to Faraday's law, this changing magnetic flux induces an emf that drives a current around the circuit.

Deriving the Motional Emf Equation

Faraday's law gives the magnitude of the induced emf as

In this example there is only one moving conductor, so [latex]N=1[/latex]. The magnetic flux is

Because the magnetic field is perpendicular to the surface enclosed by the circuit, [latex]\theta=0^\circ[/latex] and [latex]\cos\theta=1[/latex]. Therefore,

As the rod moves a distance [latex]\Delta x[/latex], it sweeps out an additional area

Substituting this expression into Faraday's law gives

Since

the induced voltage becomes

This equation shows that the induced voltage increases when:

• the magnetic field is stronger,
• the conductor is longer, or
• the conductor moves faster.

This is exactly the same expression obtained previously for the Hall effect, emphasizing that both phenomena arise from the magnetic force acting on moving electric charges.

Energy Conversion in Motional Emf

Motional emf provides a direct method for converting mechanical energy into electrical energy. As the rod moves through the magnetic field, the induced current flowing through the resistor converts some of the mechanical work used to move the rod into electrical energy, which may then be dissipated as heat or used to power another device.

This same principle is the basis of nearly every electric generator. Whether driven by a wind turbine, a hydroelectric dam, a steam turbine in a power plant, or a hand-powered generator, electrical energy is produced by moving conductors through magnetic fields or by moving magnetic fields relative to conductors.

Engineers have even attempted to exploit this effect in space. During NASA's Tethered Satellite experiments, a conducting cable several kilometers long was deployed from the Space Shuttle while orbiting Earth. As the tether moved through Earth's magnetic field at orbital speed, a large motional emf was expected to develop along its length, demonstrating that orbital motion itself can generate electrical energy.

Determining the Direction of the Induced Current

Finding the magnitude of the induced emf is only part of the problem. We must also determine the direction of the induced current and the polarity of the induced voltage. To do this, we apply Lenz's law, which states that the induced current always creates a magnetic field that opposes the change in magnetic flux.

For the situation shown in Figure 48.1(b), the conducting rod moves to the right, increasing the area enclosed by the circuit. Because the external magnetic field points into the page, the magnetic flux into the page is increasing.

According to Lenz's law, the induced magnetic field must oppose this increase in flux. Therefore, the induced field points out of the page. Applying Right-Hand Rule 2 (RHR-2), the fingers curl in the direction of the induced current while the thumb points out of the page. This shows that the induced current is counterclockwise, making the top of the rod positive and the bottom negative, exactly as shown in Figure 48.1(b).

Relative Motion Is What Matters

The conductor does not have to move for motional emf to occur. The same effect is observed whenever there is relative motion between a conductor and a magnetic field. For example, in the previous chapter we saw that moving a bar magnet toward a stationary coil induces an emf in the coil, even though the conductor itself does not move.

These examples reinforce one of the central ideas of electromagnetism: changing magnetic fields produce electric fields, while moving electric charges produce magnetic fields. Electricity and magnetism are therefore deeply connected aspects of the same physical interaction.

Motional Emf in Everyday Life

Although motional emf is always present when a conductor moves through a magnetic field, the effect is usually extremely small in Earth's relatively weak magnetic field. For example, a 1.0 m metal rod moving at 3.0 m/s perpendicular to Earth's magnetic field develops an emf of only

This voltage is far too small to notice during ordinary activities, which agrees with everyday experience.

Engineering Example: The Tethered Satellite Experiment

One ambitious attempt to take advantage of motional emf occurred during NASA's Tethered Satellite System missions in the 1990s. Engineers deployed a conducting cable intended to extend nearly 20 km from the Space Shuttle while orbiting Earth, as illustrated in Figure 48.2.

As the shuttle traveled through Earth's magnetic field at orbital speed, the long conducting tether was expected to generate an emf of approximately 5 kV. If a complete electrical circuit could be established, part of the shuttle's mechanical energy would be converted into electrical energy.

In this system, the ionosphere—the partially ionized upper atmosphere—would act as the return path for the current, completing the circuit much like the rails and resistor in Figure 48.1. As current flowed through the tether, the magnetic force

would oppose the shuttle's motion. This magnetic drag would perform work on the electrical charges, converting some of the shuttle's kinetic and gravitational potential energy into electrical energy.

Although the missions encountered technical problems and did not fully achieve their objectives—the tether jammed during one mission and broke during another—they demonstrated the feasibility of generating electrical energy directly from orbital motion through Earth's magnetic field. The following example illustrates the underlying physics of this process.

Example 48.1: Calculating the Motional Emf of a Conductor in Earth Orbit

Calculate the motional emf induced along a 20.0 km long conducting cable moving at an orbital speed of 7.80 km/s perpendicular to Earth's magnetic field of [latex]5.00\times10^{-5}\,\text{T}[/latex].

Strategy

Since the conductor moves perpendicular to the magnetic field, the induced voltage is found directly using the motional emf equation

[latex]\text{emf}=B\ell v.[/latex]

All three quantities are provided, so we simply substitute the known values.

Solution

Substituting the given values gives

[latex]\begin{aligned} \text{emf} &=B\ell v\\[4pt] &=\left(5.00\times10^{-5}\,\text{T}\right) \left(2.00\times10^{4}\,\text{m}\right) \left(7.80\times10^{3}\,\text{m/s}\right)\\[4pt] &=7.80\times10^{3}\,\text{V}. \end{aligned}[/latex]

Therefore, the maximum induced emf is

[latex]\boxed{\text{emf}=7.80\times10^{3}\,\text{V}=7.80\,\text{kV}.}[/latex]

Discussion

This result represents the maximum possible induced voltage, which occurs only when the conductor moves exactly perpendicular to the magnetic field. During the actual Tethered Satellite missions, the cable was not perfectly perpendicular to Earth's magnetic field throughout its orbit, so the predicted operating voltage was closer to 5 kV.

Although the missions experienced technical difficulties before the full experiment could be completed, the calculation illustrates an important principle: very large voltages can be generated simply by moving a long conductor rapidly through a magnetic field. This same principle is used in electric generators, although rotating coils and magnets are far more practical than deploying a 20-km cable in space.