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Chapter 25: Problem 9

Use Faraday's law to derive an expression for the potential difference across a planet by a magnetic field \(B\) sweeping across the planet's surface at a speed \(v\). Take the planet radius to be \(R\).

Short Answer

Expert verified
The potential difference is \( 2 B R v \).

Step by step solution

01

Understand Faraday's Law

Faraday's Law states that the electromotive force (EMF) induced in a loop is equal to the negative rate of change of magnetic flux through the loop. Mathematically, it is expressed as \(\text{EMF} = -\frac{d\text{Φ}}{dt}\), where \(\text{Φ}\) is the magnetic flux.
02

Define Magnetic Flux

Magnetic flux \(\text{Φ}\) through a surface is given by the product of the magnetic field \(B\) and the area \(A\) it penetrates perpendicular to the field. For a circular cross-section of radius \(R\), the flux is \(\text{Φ} = B \times A = B \times (\frac{\text{π} R^2}{2})\).
03

Determine the Change in Flux

As the magnetic field sweeps across the planet at speed \(v\), the area changes over time. The rate of change of area is the speed \(v\) times the length of the diameter \(2R\). Therefore, \(\frac{dA}{dt} = 2R \times v\).
04

Apply Faraday’s Law

Using Faraday’s Law, \(\text{EMF} = -B \frac{dA}{dt}\). Substitute \(\frac{dA}{dt} = 2R \times v\): \(\text{EMF} = -B \times (2R \times v)\).
05

Simplify the Expression

Simplifying the above expression: \( \text{EMF} = -2 B R v\). The negative sign indicates the direction of the induced EMF, which can be ignored for magnitude.

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Key Concepts

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

Magnetic Flux
Magnetic flux is a measure of the total magnetic field passing through a particular area. Imagine the magnetic field as invisible lines passing through a surface. The magnetic flux quantifies how many of these lines pass through that surface.
The magnetic flux \( \Phi \) can be calculated using the formula: \[ \Phi = B \cdot A \cdot \cos(\theta) \] In this formula:
  • \( B \) is the magnetic field strength
  • \( A \) is the area the field is passing through
  • \( \theta \) is the angle between the magnetic field and the normal (perpendicular) to the surface
For simplicity, if the magnetic field is perpendicular to the surface (\(\theta = 0\)), the equation simplifies to: \[ \Phi = B \cdot A \] In the context of our planet example, we used the radius of the planet to find the area through which the magnetic field passes.
Induced Electromotive Force
The induced electromotive force (EMF) is a voltage generated by changing magnetic flux. According to Faraday’s Law of Induction, the EMF is equal to the negative rate of change of the magnetic flux.
  • The formula is: \[ \text{EMF} = -\frac{d\Phi}{dt} \] \where \( d\Phi \) is the change in flux, and \( dt \) is the change in time.
In simpler terms, when the magnetic flux through a circuit changes, it induces a voltage (or EMF) in that circuit. The faster the change in flux, the greater the induced EMF.
In our exercise, because the magnetic field sweeps across the planet's surface at a speed \( v \), the change in area over time leads to a change in flux, thus inducing an EMF.
Planetary Magnetism
Planetary magnetism refers to a planet's magnetic field, which is generated by movements within its core. For Earth, it’s the movement of molten iron in the outer core that creates geomagnetic fields.
These fields extend into space and can interact with solar wind, creating phenomena like the auroras. When considering the effect of a magnetic field sweeping across a planet, we treat the planet as a circular cross-section where the magnetic field intersects.
  • The radius of the planet is essential in our calculations, determining the area that interacts with the magnetic field.
This intercepting area then affects the magnetic flux and, subsequently, the induced EMF.
Rate of Change of Flux
The rate of change of flux is a key component in Faraday’s Law, dictating the strength of the induced EMF. It represents how quickly the magnetic flux through a surface changes over time.
  • In our problem, we calculated it as \( \frac{dA}{dt} = 2R \times v \)\ where \( R \) is the radius of the planet and \( v \) is the speed at which the magnetic field sweeps across the planet.
The faster this ‘sweeping’ action (or the greater the velocity), the more significant the rate of change of flux, and thus, the greater the induced EMF. \( \text{EMF} = -B \times 2R \times v \) illustrates this relationship clearly, showing the dependence of induced voltage on the changing area of magnetic influence.

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Most popular questions from this chapter

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