Motional EMF

Motional EMF is the potential difference produced when a conductor moves through a magnetic field and cuts across magnetic field lines.

The effect occurs because charged particles inside the moving conductor experience a magnetic force. Positive and negative charges are pushed toward opposite ends of the conductor, producing a separation of charge and therefore a potential difference.

If the conductor is part of a complete circuit, this EMF can drive an induced current.

Motional EMF is an important example of electromagnetic induction and is one of the basic principles behind electrical generators.

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EMF

EMF stands for electromotive force.

Despite its name, EMF is not actually a force.

It is energy transferred per unit charge:

EMF = energy transferred / charge

Its unit is the:

volt (V)

An EMF provides the energy needed to move charge around a circuit.

Examples of EMF sources include:

• batteries
• generators
• solar cells
• moving conductors in magnetic fields

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Moving a Conductor Through a Magnetic Field

Consider a straight metal rod moving through a uniform magnetic field.

The metal contains mobile electrons.

As the rod moves, these charges also move through the magnetic field and experience a magnetic force.

For a charge moving through a magnetic field:

F = qvB sin θ

where:

• F = magnetic force
• q = charge
• v = speed of the charge through the field
• B = magnetic field strength
• θ = angle between the velocity and magnetic field

The magnetic force causes charges to move toward opposite ends of the rod.

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Charge Separation

Suppose a conducting rod moves through a magnetic field.

The magnetic force pushes electrons toward one end of the rod.

This produces:

• excess negative charge at one end
• excess positive charge at the other end

The charge separation creates an electric field inside the conductor.

The two ends of the rod now have different electric potentials.

Therefore, an EMF has been induced.

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Magnetic Force and Electric Force

As charges separate, an electric field develops inside the rod.

The electric force on a charge is:

F_E = qE

The magnetic force is:

F_B = qvB

for motion perpendicular to the magnetic field.

Charge separation continues until these forces balance:

qE = qvB

Cancel q:

E = vB

For a rod of length L:

EMF = EL

Therefore:

EMF = BLv

when the rod moves perpendicular to the magnetic field and its length is oriented appropriately.

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The Motional EMF Equation

For a straight conductor moving through a uniform magnetic field:

ε = BLv

when the conductor, velocity, and magnetic field are mutually perpendicular in the required directions.

where:

• ε = induced EMF in volts (V)
• B = magnetic field strength in teslas (T)
• L = effective length of conductor in metres (m)
• v = speed in metres per second (m/s)

This is the most common motional EMF equation.

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Factors Affecting Motional EMF

From:

ε = BLv

the induced EMF increases when:

• magnetic field strength increases
• conductor length increases
• conductor speed increases

Therefore:

stronger field → greater EMF

longer conductor → greater EMF

faster motion → greater EMF

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Effect of Magnetic Field Strength

If L and v remain constant:

ε ∝ B

Suppose the magnetic field doubles.

Then:

the induced EMF doubles

For example, if:

B = 0.20 T

produces:

ε = 1.5 V

then increasing the field to:

B = 0.40 T

would produce:

ε = 3.0 V

assuming all other conditions remain unchanged.

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Effect of Conductor Length

If B and v remain constant:

ε ∝ L

A longer conductor cuts across more magnetic field and develops a greater potential difference between its ends.

If the effective conductor length triples:

the motional EMF triples

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Effect of Speed

If B and L remain constant:

ε ∝ v

Moving the conductor twice as fast produces:

twice the EMF

Moving it three times as fast produces:

three times the EMF

This is an important principle in generators: increasing the rate at which conductors move through a magnetic field can increase the generated voltage.

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Angle of Motion

The largest motional EMF occurs when the conductor moves perpendicular to the magnetic field.

For more general motion:

ε = BLv sin θ

where θ is the angle between the velocity and magnetic field.

When:

θ = 90°

sin 90° = 1

so:

ε = BLv

When:

θ = 0°

sin 0° = 0

so:

ε = 0

Therefore, motion parallel to the magnetic field does not produce motional EMF in this simple arrangement.

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Example: Effect of Angle

Suppose:

B = 0.50 T

L = 0.80 m

v = 6.0 m/s

θ = 30°

Then:

ε = BLv sin θ

ε = (0.50)(0.80)(6.0)(sin 30°)

Since:

sin 30° = 0.5

ε = (0.50)(0.80)(6.0)(0.5)

ε = 1.2 V

If the conductor moved perpendicular to the field instead:

ε = (0.50)(0.80)(6.0)

ε = 2.4 V

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Maximum and Minimum Motional EMF

Maximum EMF occurs when:

v ⟂ B

because:

sin 90° = 1

Therefore:

ε_max = BLv

Zero EMF occurs when:

v ∥ B

because:

sin 0° = 0

Therefore:

ε = 0

The direction of motion matters just as much as the speed.

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Worked Example: Calculating Motional EMF

A conducting rod has length:

L = 0.60 m

It moves at:

v = 5.0 m/s

perpendicular to a magnetic field of:

B = 0.80 T

Use:

ε = BLv

ε = (0.80)(0.60)(5.0)

ε = 2.4 V

The potential difference between the ends of the rod is:

2.4 V

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Worked Example: Finding Speed

A 0.50 m conductor produces an EMF of:

3.0 V

while moving perpendicular to a:

1.2 T

magnetic field.

Use:

ε = BLv

Rearrange:

v = ε/(BL)

Substitute:

v = 3.0/[(1.2)(0.50)]

v = 3.0/0.60

v = 5.0 m/s

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Worked Example: Finding Magnetic Field Strength

A 0.40 m rod moves at:

8.0 m/s

and produces:

1.6 V

of motional EMF.

Use:

ε = BLv

Rearrange:

B = ε/(Lv)

B = 1.6/[(0.40)(8.0)]

B = 1.6/3.2

B = 0.50 T

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Open Circuit and Closed Circuit

Motional EMF can exist even if the circuit is not complete.

If the rod is moving through the field but the circuit is open:

• charges separate
• a potential difference develops
• no continuous current flows

If the circuit is closed:

• the induced EMF drives charge around the circuit
• an induced current flows

Therefore:

EMF does not automatically mean current.

A complete conducting path is required for continuous current.

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Induced Current

If a motional EMF is produced in a closed circuit, the current can be calculated using Ohm's law:

I = ε/R

where:

• I = current in amperes
• ε = induced EMF in volts
• R = total circuit resistance in ohms

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Example: Induced Current

A rod produces:

ε = 2.4 V

The total circuit resistance is:

R = 6.0 Ω

Then:

I = ε/R

I = 2.4/6.0

I = 0.40 A

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Direction of the Magnetic Force

The direction of the induced EMF depends on the direction in which charges are pushed.

For a positive charge, the magnetic force direction follows:

F = q(v × B)

The directions of:

• motion
• magnetic field
• magnetic force

are mutually perpendicular when the effect is maximum.

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Right-Hand Rule

A right-hand rule can be used for positive charges.

One useful version is:

• point your fingers in the direction of the positive charge's velocity
• curl them toward the magnetic field
• your thumb indicates the magnetic force on a positive charge

For electrons, the force is in the opposite direction because electrons have negative charge.

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Field Into and Out of the Page

Magnetic-field diagrams often represent a field perpendicular to the page.

A field out of the page is commonly shown using dots:

•

Think of seeing the point of an arrow coming toward you.

A field into the page is commonly shown using crosses:

×

Think of seeing the tail feathers of an arrow moving away from you.

These symbols are frequently used in motional EMF problems.

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Example: Direction of Charge Separation

Suppose:

• a vertical conducting rod moves to the right
• the magnetic field points into the page

For a positive charge:

• velocity is right
• magnetic field is into the page
• magnetic force is upward

Positive charge therefore accumulates toward the upper end.

Electrons experience the opposite force and move toward the lower end.

Therefore:

top of rod → higher potential

bottom of rod → lower potential

If the rod forms part of a closed circuit, this polarity helps determine the direction of conventional current.

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Conventional Current and Electron Flow

Conventional current is defined as the direction positive charge would move.

Electron flow is opposite to conventional current.

Therefore, when using the right-hand rule:

• first determine the force on a positive charge
• use that direction for conventional current
• reverse it if specifically asked for electron motion

This avoids a common source of confusion.

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Reversing the Motion

If the direction of motion reverses while the magnetic field remains unchanged, the magnetic force reverses.

Therefore:

• charge separation reverses
• polarity reverses
• induced current reverses

So changing:

v → −v

reverses the direction of the induced EMF.

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Reversing the Magnetic Field

If the magnetic field reverses while motion remains unchanged:

• magnetic force reverses
• charge separation reverses
• induced current reverses

Therefore, reversing either the velocity or magnetic field reverses the induced current.

Reversing both causes two reversals, so the original current direction is restored.

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Sliding Rod on Conducting Rails

A common motional EMF arrangement consists of:

• two parallel conducting rails
• a conducting rod connecting the rails
• a magnetic field perpendicular to the plane of the circuit
• the rod sliding along the rails

As the rod moves, the area of the circuit changes.

A motional EMF is produced in the moving rod.

If the circuit is complete, an induced current flows.

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Connection to Magnetic Flux

Magnetic flux through a simple uniform-field surface can be written:

Φ = BA cos θ

where θ is the angle between the magnetic field and the normal to the surface.

For a sliding rod, the area enclosed by the circuit changes as the rod moves.

If:

A = Lx

then:

Φ = BLx

for a field perpendicular to the loop.

Since:

v = Δx/Δt

the rate of flux change has magnitude:

ΔΦ/Δt = BLv

Faraday's law gives an induced EMF with magnitude:

ε = BLv

This shows that motional EMF is consistent with the broader principle of electromagnetic induction.

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Faraday's Law

Faraday's law states that an EMF is induced when magnetic flux through a circuit changes.

In magnitude:

|ε| = |ΔΦ/Δt|

More generally:

ε = −dΦ/dt

The negative sign represents Lenz's law.

Motional EMF is therefore one way of producing a changing magnetic flux.

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Lenz's Law

Lenz's law states that the induced current acts in a direction that opposes the change that produced it.

For a sliding rod, the induced current interacts with the magnetic field to produce a magnetic force that opposes the rod's motion.

This does not mean the rod must stop.

It means an external force must do work to keep the rod moving at constant speed.

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Magnetic Braking Force

Suppose a rod moves along rails and produces an induced current.

The current-carrying rod is now inside a magnetic field.

A magnetic force acts on it.

The direction of this force opposes the motion responsible for the induction.

Therefore:

motion → induced EMF → induced current → opposing magnetic force

This is electromagnetic braking.

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Energy Conservation

Motional EMF does not create electrical energy from nothing.

If an external force pushes the rod through the magnetic field, that force does mechanical work.

Energy is transferred from:

mechanical energy → electrical energy

If the circuit contains resistance, much of that electrical energy eventually becomes:

thermal energy

This is consistent with conservation of energy.

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Worked Example: Rod and Rails

A conducting rod has:

L = 0.50 m

and moves at:

v = 4.0 m/s

through a magnetic field:

B = 0.60 T

The circuit resistance is:

R = 3.0 Ω

First calculate EMF:

ε = BLv

ε = (0.60)(0.50)(4.0)

ε = 1.2 V

Now calculate current:

I = ε/R

I = 1.2/3.0

I = 0.40 A

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Electrical Power

The electrical power delivered to a resistive circuit is:

P = Iε

It can also be written:

P = I²R

or:

P = ε²/R

Using the previous example:

P = Iε

P = (0.40)(1.2)

P = 0.48 W

This energy ultimately comes from the mechanical work required to move the rod.

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Mechanical and Electrical Power

If the rod moves at constant speed, the external force must balance the opposing magnetic force.

Mechanical power supplied is:

P = Fv

For an ideal system:

mechanical power input = electrical power output

This provides another demonstration of energy conservation.

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Generators

Electrical generators use electromagnetic induction to convert:

mechanical energy → electrical energy

Rather than simply moving a straight rod through a field, many generators rotate conducting coils in magnetic fields.

The conductors continuously move relative to the magnetic field.

This produces an induced EMF.

Sources of mechanical motion may include:

• steam turbines
• wind turbines
• water turbines
• engines
• human motion

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Bicycle Generators

Some bicycle lighting systems use a small generator.

As the bicycle moves:

• part of the generator rotates
• conductors experience changing magnetic conditions
• an EMF is induced
• current powers the light

The energy ultimately comes from the cyclist's mechanical work.

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Moving Wires in Magnetic Fields

Motional EMF can occur whenever a conductor moves through a magnetic field with an appropriate component of velocity perpendicular to the field.

Examples include:

• moving rods
• rotating conducting loops
• generator coils
• rotating disks
• conducting components in machinery

The exact geometry may change, but the underlying idea remains the same:

motion of charges through a magnetic field produces magnetic forces and charge separation.

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Microphones and Electromagnetic Pickups

Some devices use electromagnetic induction caused by motion.

In a dynamic microphone, motion caused by sound moves a coil relative to a magnetic field, producing an electrical signal.

Electric guitar pickups use a related induction process. Vibrating steel strings alter the magnetic flux through nearby coils, inducing a voltage.

These devices involve electromagnetic induction, although their detailed operation is more complex than the simple straight-rod motional EMF model.

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Aircraft and Moving Conductors

A conducting object moving through Earth's magnetic field can develop a very small motional EMF.

For example, a conducting aircraft structure moving through a magnetic field can experience charge separation.

The effect is usually small because Earth's magnetic field is weak, but it illustrates that motional EMF is not limited to laboratory magnets.

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Example: Aircraft Wing

Suppose a conducting wing has an effective length:

L = 30 m

moves at:

v = 250 m/s

perpendicular to an effective magnetic field component:

B = 4.0 × 10⁻⁵ T

Then:

ε = BLv

ε = (4.0 × 10⁻⁵)(30)(250)

ε = 0.30 V

Even a weak magnetic field can produce a measurable EMF if the conductor is long and moving quickly.

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Selecting the Correct Equation

Use:

ε = BLv

when:

• a straight conductor moves through a uniform magnetic field
• the motion is perpendicular to the field
• the effective conductor length is appropriately oriented

Use:

ε = BLv sin θ

when the velocity is at an angle θ to the magnetic field in the standard simple geometry.

Use:

I = ε/R

when the circuit is complete and resistance is known.

Use Faraday's law when the problem is more naturally described using changing magnetic flux.

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A Complete Calculation Problem

A 0.75 m conducting rod moves at:

6.0 m/s

through a uniform magnetic field:

B = 0.40 T

The velocity is perpendicular to the field.

The total circuit resistance is:

2.0 Ω

Calculate the EMF:

ε = BLv

ε = (0.40)(0.75)(6.0)

ε = 1.8 V

Calculate current:

I = ε/R

I = 1.8/2.0

I = 0.90 A

Calculate electrical power:

P = Iε

P = (0.90)(1.8)

P = 1.62 W

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A Complete Direction Problem

A vertical conducting rod moves to the left through a magnetic field directed into the page.

For a positive charge:

• velocity points left
• magnetic field points into the page
• right-hand rule gives force downward

Therefore:

• positive charge accumulates at the bottom
• electrons accumulate toward the top

The bottom of the rod is therefore at higher electric potential than the top.

If the rod is part of a closed circuit, conventional current leaves the higher-potential end through the external circuit.

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A Reliable Strategy for Motional EMF Problems

First identify:

• magnetic field direction
• direction of motion
• conductor length
• speed
• whether the circuit is open or closed

For magnitude, determine whether the motion is perpendicular to the field.

If perpendicular:

ε = BLv

If an angle is involved:

ε = BLv sin θ

For current:

I = ε/R

For direction:

1. identify v
2. identify B
3. apply the right-hand rule to a positive charge
4. determine which end becomes positive
5. determine conventional current direction if the circuit is closed

Finally, check whether the answer makes physical sense.

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Checking Your Answer

Ask:

• Does increasing B increase the EMF?
• Does increasing L increase the EMF?
• Does increasing v increase the EMF?
• Is the EMF zero if motion is parallel to B?
• Does reversing motion reverse the polarity?
• Does reversing B reverse the polarity?
• Is there a complete circuit if current is being calculated?
• Does the direction agree with the right-hand rule?
• Is the energy ultimately coming from mechanical work?

These checks can catch both calculation and reasoning errors.

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Common Mistakes

Thinking EMF is a force

EMF is measured in volts and represents energy transferred per unit charge.

Assuming any motion produces an EMF

The motion must have an appropriate component across the magnetic field.

Forgetting the angle

If motion is not perpendicular to B, the perpendicular component must be considered.

Using ε = BLv when the conductor moves parallel to B

If v is parallel to B:

ε = 0

in the simple rod model.

Confusing conventional current with electron flow

They are opposite directions.

Applying the right-hand rule directly to electrons

The standard magnetic-force right-hand rule gives the force direction for a positive charge. Reverse it for electrons.

Thinking an open circuit cannot have an EMF

An open circuit can have a potential difference even though continuous current does not flow.

Calculating current without resistance

Current depends on both EMF and circuit resistance.

Forgetting Lenz's law

The induced current produces effects that oppose the change responsible for the induction.

Assuming electrical energy appears without an energy source

Mechanical work provides the energy in a motional EMF system.

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Key Terms

Motional EMF — potential difference produced when a conductor moves through a magnetic field.

Electromotive force (EMF) — energy transferred per unit charge, measured in volts.

Magnetic field — region in which magnetic forces can act on moving charges or magnetic materials.

Magnetic force — force experienced by a moving charge in a magnetic field.

Charge separation — accumulation of opposite charges at different locations within a conductor.

Induced EMF — EMF produced by electromagnetic induction.

Induced current — current produced by an induced EMF in a complete circuit.

Conventional current — direction in which positive charge is defined to move.

Electron flow — movement of electrons, opposite to conventional current.

Right-hand rule — method used to determine directions involving velocity, magnetic field, and magnetic force for positive charge.

Magnetic flux — measure related to the magnetic field passing through a surface.

Faraday's law — principle relating induced EMF to the rate of change of magnetic flux.

Lenz's law — principle stating that induced effects oppose the change that produced them.

Generator — device that converts mechanical energy into electrical energy using electromagnetic induction.

Open circuit — incomplete circuit in which continuous current cannot flow.

Closed circuit — complete conducting path through which current can flow.

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Key Takeaways

• Motional EMF occurs when a conductor moves appropriately through a magnetic field.
• Moving charges in the conductor experience a magnetic force.
• The magnetic force separates charges and creates a potential difference.
• For perpendicular motion in the standard straight-rod arrangement, ε = BLv.
• More generally, the relevant perpendicular motion gives ε = BLv sin θ in the simple geometry.
• Increasing magnetic field strength increases motional EMF.
• Increasing effective conductor length increases motional EMF.
• Increasing speed increases motional EMF.
• Maximum EMF occurs when motion is perpendicular to the magnetic field.
• Motion parallel to the magnetic field produces zero motional EMF in the simple model.
• An EMF can exist in an open circuit even when no continuous current flows.
• In a closed circuit, the induced current can be calculated using I = ε/R.
• The right-hand rule determines magnetic force direction for positive charges.
• Electron motion is opposite to the force direction determined for positive charge.
• Reversing either the motion or magnetic field reverses the induced polarity and current.
• Motional EMF can also be understood using changing magnetic flux and Faraday's law.
• Lenz's law explains why induced magnetic effects oppose the change producing them.
• Mechanical work is converted into electrical energy.
• Generators are important practical applications of electromagnetic induction.
• Motional EMF connects magnetic forces, electromagnetic induction, energy conservation, and electrical generation.

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Check Your Understanding

1. Explain how moving a conducting rod through a magnetic field can create a potential difference between its ends.
2. What three quantities determine the magnitude of ε = BLv for perpendicular motion?
3. A 0.50 m rod moves at 4.0 m/s perpendicular to a 0.60 T magnetic field. Calculate the induced EMF.
4. A 0.80 m conductor moves at 10 m/s through a 0.25 T field. Calculate the maximum motional EMF.
5. A conductor produces an EMF of 4.0 V while moving at 5.0 m/s through a 0.80 T field. Determine its effective length.
6. A 0.40 m conductor moves through a 0.50 T field and produces an EMF of 1.2 V. Determine its speed.
7. A rod moves at 8.0 m/s at an angle of 30° to a 0.60 T magnetic field. Its effective length is 0.50 m. Calculate the motional EMF.
8. Explain why a rod moving parallel to a magnetic field produces no motional EMF in the simple model.
9. A rod produces 3.0 V of motional EMF in a circuit with resistance 5.0 Ω. Calculate the induced current.
10. A vertical rod moves to the right while the magnetic field points into the page. Use the right-hand rule to determine which end of the rod becomes positive.
11. What happens to the direction of the induced current if the rod's direction of motion is reversed? Explain.
12. What happens if both the magnetic field and direction of motion are reversed simultaneously?
13. Explain why the magnetic force produced by the induced current tends to oppose the motion of a sliding rod.
14. A 0.60 m rod moves at 5.0 m/s through a 0.75 T field. The circuit resistance is 3.0 Ω. Calculate the induced EMF, current, and electrical power.
15. Explain how motional EMF demonstrates the conversion of mechanical energy into electrical energy, and describe one practical device in which electromagnetic induction is used for this purpose.