# 23.3 Motional emf  (Page 3/3)

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## Section summary

• An emf induced by motion relative to a magnetic field $B$ is called a motional emf and is given by
$\text{emf}=\mathrm{B\ell v}\phantom{\rule{3.00em}{0ex}}\text{(}B\text{,}\phantom{\rule{0.25em}{0ex}}\ell \text{, and}\phantom{\rule{0.25em}{0ex}}v\phantom{\rule{0.25em}{0ex}}\text{perpendicular),}$
where $\ell$ is the length of the object moving at speed $v$ relative to the field.

## Conceptual questions

Why must part of the circuit be moving relative to other parts, to have usable motional emf? Consider, for example, that the rails in [link] are stationary relative to the magnetic field, while the rod moves.

A powerful induction cannon can be made by placing a metal cylinder inside a solenoid coil. The cylinder is forcefully expelled when solenoid current is turned on rapidly. Use Faraday’s and Lenz’s laws to explain how this works. Why might the cylinder get live/hot when the cannon is fired?

An induction stove heats a pot with a coil carrying an alternating current located beneath the pot (and without a hot surface). Can the stove surface be a conductor? Why won’t a coil carrying a direct current work?

Explain how you could thaw out a frozen water pipe by wrapping a coil carrying an alternating current around it. Does it matter whether or not the pipe is a conductor? Explain.

## Problems&Exercises

Use Faraday’s law, Lenz’s law, and RHR-1 to show that the magnetic force on the current in the moving rod in [link] is in the opposite direction of its velocity.

If a current flows in the Satellite Tether shown in [link] , use Faraday’s law, Lenz’s law, and RHR-1 to show that there is a magnetic force on the tether in the direction opposite to its velocity.

(a) A jet airplane with a 75.0 m wingspan is flying at 280 m/s. What emf is induced between wing tips if the vertical component of the Earth’s field is $3\text{.}\text{00}×{\text{10}}^{-5}\phantom{\rule{0.25em}{0ex}}\text{T}$ ? (b) Is an emf of this magnitude likely to have any consequences? Explain.

(a) 0.630 V

(b) No, this is a very small emf.

(a) A nonferrous screwdriver is being used in a 2.00 T magnetic field. What maximum emf can be induced along its 12.0 cm length when it moves at 6.00 m/s? (b) Is it likely that this emf will have any consequences or even be noticed?

At what speed must the sliding rod in [link] move to produce an emf of 1.00 V in a 1.50 T field, given the rod’s length is 30.0 cm?

2.22 m/s

The 12.0 cm long rod in [link] moves at 4.00 m/s. What is the strength of the magnetic field if a 95.0 V emf is induced?

Prove that when $B$ , $\ell$ , and $v$ are not mutually perpendicular, motional emf is given by $\text{emf}=\mathrm{B\ell v}\phantom{\rule{0.25}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ . If $v$ is perpendicular to $B$ , then $\theta$ is the angle between $\ell$ and $B$ . If $\ell$ is perpendicular to $B$ , then $\theta$ is the angle between $v$ and $B$ .

In the August 1992 space shuttle flight, only 250 m of the conducting tether considered in [link] could be let out. A 40.0 V motional emf was generated in the Earth’s $5\text{.}\text{00}×{\text{10}}^{-5}\phantom{\rule{0.25em}{0ex}}\text{T}$ field, while moving at $7\text{.}\text{80}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ . What was the angle between the shuttle’s velocity and the Earth’s field, assuming the conductor was perpendicular to the field?

Integrated Concepts

Derive an expression for the current in a system like that in [link] , under the following conditions. The resistance between the rails is $R$ , the rails and the moving rod are identical in cross section $A$ and have the same resistivity $\rho$ . The distance between the rails is l, and the rod moves at constant speed $v$ perpendicular to the uniform field $B$ . At time zero, the moving rod is next to the resistance $R$ .

Integrated Concepts

The Tethered Satellite in [link] has a mass of 525 kg and is at the end of a 20.0 km long, 2.50 mm diameter cable with the tensile strength of steel. (a) How much does the cable stretch if a 100 N force is exerted to pull the satellite in? (Assume the satellite and shuttle are at the same altitude above the Earth.) (b) What is the effective force constant of the cable? (c) How much energy is stored in it when stretched by the 100 N force?

Integrated Concepts

The Tethered Satellite discussed in this module is producing 5.00 kV, and a current of 10.0 A flows. (a) What magnetic drag force does this produce if the system is moving at 7.80 km/s? (b) How much kinetic energy is removed from the system in 1.00 h, neglecting any change in altitude or velocity during that time? (c) What is the change in velocity if the mass of the system is 100,000 kg? (d) Discuss the long term consequences (say, a week-long mission) on the space shuttle’s orbit, noting what effect a decrease in velocity has and assessing the magnitude of the effect.

(a) 10.0 N

(b) $2.81×{\text{10}}^{\text{8}}\phantom{\rule{0.25em}{0ex}}\text{J}$

(c) 0.36 m/s

(d) For a week-long mission (168 hours), the change in velocity will be 60 m/s, or approximately 1%. In general, a decrease in velocity would cause the orbit to start spiraling inward because the velocity would no longer be sufficient to keep the circular orbit. The long-term consequences are that the shuttle would require a little more fuel to maintain the desired speed, otherwise the orbit would spiral slightly inward.

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