91Ó°ÊÓ

A flat coil of copper wire consists of 100 loops and has a total resistance of \(0.500 \Omega\). The coil diameter is \(4.00 \mathrm{~cm}\) and it is in a uniform magnetic field pointing toward you (out the page). The coil orientation is in the plane of the page. It is then pulled to the right (without rotating) until it is completely out of the field. (a) What is the direction of the induced current in the coil: (1) clockwise, (2) counterclockwise, or (3) there is no induced current? (b) During the time the coil leaves the field, an average induced current of \(20.0 \mathrm{~mA}\) is measured. What is the average induced emf in the coil? (c) If the field strength is \(5.50 \mathrm{mT}\), how much time did it take to pull the coil out?

Step by step solution

01

Understanding the Direction of Induced Current

To determine the direction of the induced current, we apply Lenz's Law, which states that the direction of induced current will oppose the change in magnetic flux. Since the coil is pulled out of a uniform magnetic field, the flux through the coil decreases. Therefore, the induced current must create a magnetic field pointing out of the page. By the right-hand rule, this requires a counterclockwise current.

02

Calculating the Average Induced EMF

Using Ohm's Law, the relationship between induced emf \( \varepsilon \), current \( I \), and resistance \( R \) is given by \( \varepsilon = I \times R \). Given that \( I = 20.0 \, \text{mA} = 0.020 \, \text{A} \) and \( R = 0.500 \, \Omega \), we can calculate the average induced emf as: \[ \varepsilon = 0.020 \, \text{A} \times 0.500 \, \Omega = 0.010 \, \text{V} \]

03

Finding the Time to Pull the Coil Out

The emf is also related to the change in magnetic flux \( \Delta \Phi \) over time \( \Delta t \) by \( \varepsilon = \frac{\Delta \Phi}{\Delta t} \). The total initial flux \( \Phi_{i} \) is \( B \times A \) where \( B = 5.50 \, \text{mT} = 5.50 \times 10^{-3} \, \text{T} \) and \( A = \pi \left( \frac{0.04}{2} \right)^2 \). Thus, \( \Phi_{i} = 5.50 \times 10^{-3} \, \text{T} \times \pi \left( 0.02 \, \text{m} \right)^2 \). Solve for \( \Delta t \): \[ \varepsilon = \frac{\Phi_{i}}{\Delta t} \Rightarrow \Delta t = \frac{\Phi_{i}}{\varepsilon} = \frac{5.50 \times 10^{-3} \times \pi \times 0.02^2}{0.010} \] Calculating gives \( \Delta t = 6.92 \, \text{s} \).

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Key Concepts

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

Lenz's Law

Lenz's Law is a fundamental principle that helps us determine the direction of an induced current in a coil exposed to changing magnetic environments.
It states that the induced current will flow in such a way that it oposes the change in magnetic flux that produced it. In simple terms, it tries to "resist" or "fight back" against the change.
When analyzing the situation of a coil being pulled out of a magnetic field, Lenz's Law tells us that the reduced magnetic flux triggers the coil to generate a current.

• The generated current aims to supplement the decreasing magnetic field.
• This results in the creation of a new magnetic field, acting in the direction opposite to the initial change, similar to a guardian against flux changes.

To find the direction of this induced current, one can use the right-hand rule. For this exercise, the reduced flux out of the page necessitated an opposing field out of the page, hinting at a counterclockwise current formation.

Ohm's Law

Ohm's Law is a key concept to understand the relationship between voltage (induced electromotive force, or EMF), current, and resistance in an electrical circuit.
This law is succinctly expressed by the equation:\[V = I \times R\]Where:

• \(V\) represents the voltage or induced EMF.
• \(I\) is the current in amperes.
• \(R\) stands for resistance in ohms.

In the context of the exercise at hand, we know the current through the coil is 20.0 mA, and the total resistance is 0.500 Ω.
Using the formula, we can calculate the average induced EMF in the coil:\[\varepsilon = 0.020 \times 0.500 = 0.010 \, \text{V}\]This relationship highlights how an electrical system responds to the various forces acting upon it, providing a straightforward way to calculate the potential difference across a resistor.

Magnetic Flux

Magnetic Flux refers to the total magnetic field passing through a given area. Think of it as the amount of "magnetic field lines" flowing through the loop.
This concept helps us understand how strong or weak the magnetic connection is within a particular region.Mathematically, magnetic flux \(Φ\) is calculated using the equation:\[\Phi = B \times A \times \cos(\theta)\]Where:

• \(B\) is the magnetic field strength in teslas.
• \(A\) is the area perpendicular to the magnetic field.
• \(\theta\) is the angle between the field lines and the normal to the area.

In our problem's scenario, since the coil is in the plane of the magnetic field, \(\theta\) is 0, and \(\cos(0) = 1\).
This simplifies our flux equation to:\[\Phi = B \times A\]Recognizing the magnetic flux aids in identifying how the changing field properties dictate the induced EMF.

Induced EMF

Induced Electromotive Force (EMF) is the voltage created when a conductor is exposed to a changing magnetic field.
It's the driving force for the induced current within coils or loops in such conditions.The concept of induced EMF can be expressed by the equation:\[\varepsilon = \frac{\Delta \Phi}{\Delta t}\]This equation tells us that the EMF is proportional to the rate at which magnetic flux changes over time:

• The greater the change in flux, the stronger the induced EMF.
• Conversely, if the flux change is slow, the induced EMF is weaker.

For example, in practice, as coils exit a magnetic field, the flux reduces, producing a measurable EMF.
Using our exercise's specifics, where the EMF is calculated and the initial flux determined by the strength and area, we can calculate how long the coil takes to exit the zone, which is a clear realization of this principle.