91Ó°ÊÓ

If you secure a refrigerator magnet about \(2 \mathrm{~mm}\) from the metallic surface of a refrigerator door and then move the magnet sideways, you can feel a resistive force, indicating the presence of eddy currents in the surface. (a) Estimate the magnetic field strength \(B\) of the magnet to be \(5 \mathrm{mT}\) (Problem 28.53 ) and assume the magnet is rectangular with dimensions \(4 \mathrm{~cm}\) wide by \(2 \mathrm{~cm}\) high, so its area \(A\) is \(8 \mathrm{~cm}^{2}\). Now estimate the magnetic flux \(\Phi_{B}\) into the refrigerator door behind the magnet. (b) If you move the magnet sideways at a speed of \(2 \mathrm{~cm} / \mathrm{s},\) what is a corresponding estimate of the time rate at which the magnetic flux through an area \(A\) fixed on the refrigerator is changing as the magnet passes over? Use this estimate to estimate the emf induced under the rectangle on the door's surface.

Step by step solution

01

Calculate Magnetic Flux

The magnetic flux (\(\Phi_{B}\)) is the product of the magnetic field strength and the area. In this case, \(B = 5 \, \text{mT} = 5 \times 10^{-3} \, \text{T}\) and \(A = 8 \, \text{cm}^{2} = 8 \times 10^{-4} \, m^{2}\). Thus, the flux can be calculated as follows: \(\Phi_{B} = B \times A = 5 \times 10^{-3} \, \text{T} \times 8 \times 10^{-4} \, \text{m}^{2} \).

02

Find the Rate of Change in Magnetic Flux

We assume that the entire area of the magnet sweeps past a given spot in 2 cm or \(0.02 \, \text{m}\), which takes 1 second. Hence, the flux falls from its maximum to zero in this time, \(1 \, \text{s}\). The rate of change in magnetic flux is the change in flux (\(\Delta \Phi_{B}\)) divided by the change in time (\(\Delta t\)), expressed as \(\frac{\Delta \Phi_{B}}{\Delta t}\), where \(\Delta \Phi_{B} = \Phi_{B}\) (from Step 1) and \(\Delta t = 1 \, \text{s}\) in this case.

03

Estimate the Induced Electromotive Force (emf)

The emf (\(\varepsilon\)) can be found using Faraday's law, which states that the absolute value of the change in the magnetic field equals to the induced emf. Mathematically, it comprises the negative rate of change in magnetic flux, i.e., \( \varepsilon = -\frac{\Delta \Phi_{B}}{\Delta t}\). Since we are interested in the magnitude of emf, we can ignore the negative sign, which indicates the direction of current.

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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 core concept when understanding electromagnetic interactions. It is a measure of the total magnetic field passing through a given area. Think of it as how many magnetic field lines pass through a surface. In our exercise with the refrigerator magnet:

• The formula for magnetic flux (\( \Phi_{B} \)) is calculated as: \[ \Phi_{B} = B \times A \] where \( B \) is the magnetic field strength and \( A \) is the area.
• Given the magnetic field strength \( B = 5 imes 10^{-3} \, \text{T} \) and area \( A = 8 imes 10^{-4} \, \text{m}^{2} \), we can find the magnetic flux as \( \Phi_{B} = 5 imes 10^{-3} \, \text{T} \times 8 \times 10^{-4} \, \text{m}^{2} \).

In simple terms, the magnetic flux helps us understand the strength of the magnetic interaction at play, especially as the magnet moves.

Faraday's Law

Faraday's Law is instrumental in linking magnetic flux to electromotive force (emf). This law outlines how a change in magnetic flux induces an emf in a closed loop or conductive material.
According to Faraday's Law:

• The induced emf (\( \varepsilon \)) is proportional to the rate of change of magnetic flux through a loop.
• Mathematically, it's expressed as: \[ \varepsilon = -\frac{d\Phi_{B}}{dt} \]

During our exercise, as the magnet moves across the refrigerator door, the magnetic flux through a stationary area changes over time, allowing us to calculate the induced emf across that area.
By grasping Faraday's concept, we can understand how moving magnetic fields generate electrical currents and the principles behind many electrical technologies, like generators and transformers.

Electromotive Force (emf)

The notion of electromotive force, often referred to as emf, is pivotal in understanding how electricity is generated by moving magnetic fields. Emf represents the energy provided per charge that moves through a circuit or conductive path. In simpler terms, it's what pushes electrons around a circuit.
Using Faraday's Law, we computed the emf induced by the movement of our magnet:

• As the magnet causes the magnetic flux to decrease from a maximum to zero, the rate of change yields the emf.
• The calculation of emf involves the formula: \[ \varepsilon = -\frac{\Delta \Phi_{B}}{\Delta t} \]
• Even though the negative sign in Faraday's equation indicates the direction of the induced current, we often focus on the magnitude of emf in problems, just like with the magnet example.

Understanding emf helps to illustrate how moving a magnetic source can generate electric currents without direct contact, laying groundwork for modern electric generators and induction motors.