91Ó°ÊÓ

Motional emfs in Transportation. Airplanes and trains move through the earth's magnetic field at rather high speeds, so it is reasonable to wonder whether this field can have a substantial effect on them. We shall use a typical value of 0.50 G for the earth's field (a) The French TGV train and the Japanese "bullet train" reach speeds of up to 180 mph moving on tracks about 1.5 \(\mathrm{m}\) apart. At top speed moving perpendicular to the earth's magnetic field, what potential difference is induced across the tracks as the wheels roll? Does this seem large enough to produce noticeable effects? (b) The Boeing \(747-400\) aircraft has a wingspan of 64.4 \(\mathrm{m}\) and a cruising speed of 565 mph. If there is no wind blowing (so that this is also their speed relative to the ground), what is the maximum potential difference that could be induced between the opposite tips of the wings? Does this seem large enough to cause problems with the plane?

Step by step solution

01

Convert Units

Convert the speeds from miles per hour (mph) to meters per second (m/s). Use the conversion factor 1 mph = 0.44704 m/s.- For the train traveling at 180 mph: \[v = 180 \times 0.44704 = 80.4672\, \text{m/s}\]- For the aircraft traveling at 565 mph:\[v = 565 \times 0.44704 \approx 252.8641\, \text{m/s}\]

02

Calculate Induced EMF for Train

The formula for the motional EMF (electromotive force) is given by:\[\mathcal{E} = B \cdot v \cdot L \]Where:- \( B \) is the magnetic field (`0.50 G` = `5 \times 10^{-5}` Tesla)- \( v \) is the speed of the train- \( L \) is the separation between the tracks (1.5 m)Substitute the values into the formula:\[\mathcal{E} = 5 \times 10^{-5} \times 80.4672 \times 1.5 \approx 6.035\, \text{mV}\]The induced potential difference across the tracks is approximately 6.04 mV.

03

Evaluate Train's Induced EMF Effect

The induced potential difference of approximately 6.04 mV is quite small. This is not large enough to produce noticeable effects or cause significant issues for the train.

04

Calculate Induced EMF for Aircraft

Using the same formula for the induced EMF:\[\mathcal{E} = B \cdot v \cdot L\]Where:- \( B = 5 \times 10^{-5} \) Tesla- \( v \approx 252.8641 \) m/s (calculated from the aircraft speed)- \( L = 64.4 \) m (wingspan)Substitute the values into the formula:\[\mathcal{E} = 5 \times 10^{-5} \times 252.8641 \times 64.4 \approx 0.814\, \text{V}\]The maximum potential difference induced between the wingtips is approximately 0.81 V.

05

Evaluate Aircraft's Induced EMF Effect

The induced EMF of approximately 0.81 V is also relatively small when considering the overall electrical systems on an aircraft. This value is unlikely to cause any interruption or notable problems in the Boeing 747-400's operation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electromotive Force (EMF)

Electromotive force, commonly abbreviated as EMF, is a crucial concept in understanding how electrical energy is generated. Though it sounds like a force, it is more accurately a potential difference that exists without any current flowing.

EMF is a measure of the energy provided per unit charge as the charge completes a circuit. It's not a force in the traditional sense but rather a cause that produces and maintains a current. In simple terms, EMF is what pushes electrons around a circuit.

• EMF is measured in volts (V).
• It can be induced by changing magnetic fields, as discovered by Michael Faraday.
• Sources of EMF include batteries and generators.

In our exercise, EMF is induced in a train and an airplane as they move through the earth's natural magnetic field, leading to a potential difference across their structures.

Magnetic Field

The magnetic field is an invisible field that exerts a force on charges moving through it. It is represented by the symbol 'B' and measured in Tesla (T). Magnetic fields are created by moving electric charges, such as the flow of electrons.

Earth's magnetic field is a key example, which is relatively weak but present everywhere on the surface of the earth. This magnetic field can interact with moving conductors, like the train tracks or wingspan of an aircraft described in the exercise.

• The uniform magnetic field across a region means equal intensity and direction.
• Earth's magnetic field strength is about 0.5 Gauss, which is equivalent to 5 × 10^-5 Tesla.

The magnetic field can produce noticeable effects, like the induced EMF, when an object moves through it, provided there is a component of motion perpendicular to the field.

Induced Potential Difference

The induced potential difference is created when there is a movement of a conductor in a magnetic field. This phenomenon is a fundamental principle of electromagnetic induction, a key discovery made by Faraday.

The potential difference, or voltage, appears between two points, such as the train wheels or airplane wingtips, because of the relative motion between a conductor and the magnetic field. This is reflected by the formula for motional EMF: \(\mathcal{E} = B \cdot v \cdot L\).

• \(B\) - Magnetic field strength in Tesla.
• \(v\) - Speed of the conductor perpendicularly.
• \(L\) - Length of the conductor.

In simple terms, as the train or plane moves, it cuts through magnetic field lines, inducing a voltage, though usually quite small relative to other electronics systems onboard.

Lenz's Law

Lenz's Law is essential in understanding the direction of induced EMF and current. This law states that the direction of the induced EMF will always oppose the change that causes it. It's a consequence of the law of conservation of energy.

Whenever a conductor moves through a magnetic field, the induced current will create its own magnetic field. This induced field acts in opposition to the original change, trying to maintain the status quo.

• If a conductor is moved through a magnetic field, a current is induced in such a way that it opposes the movement.
• This can help predict the behavior of the system under the influence of change.

In practical terms, in our exercise, Lenz's Law helps understand that the electrical effects induced in the train or plane are self-regulatory, preventing any runaway currents from developing.