Problem 10 - University Physics with Modern Physics

Question

A rectangle measuring 30.0 \(\mathrm{cm}\) by 40.0 \(\mathrm{cm}\) is located inside a region of a spatially uniform magnetic field of 1.25 \(\mathrm{T}\) , with the field perpendicular to the plane of the coil (Fig. 29.29 ). The coil is pulled out at a steady rate of 2.00 \(\mathrm{cm} / \mathrm{s}\) traveling perpendicular to the field lines. The region of the field ends abruptly as shown. Find the emf induced in this coil when it is (a) all inside the field: (b) partly inside the field; (c) all outside the field.

Step-by-Step Solution

Answer

(a) 0 V; (b) -75.0 mV; (c) 0 V.

Step 1: Understanding the Scenario

We have a rectangle coil moving through a magnetic field. The coil dimensions are 30.0 cm by 40.0 cm, and it moves perpendicular to the uniform magnetic field of 1.25 T at a speed of 2.00 cm/s. We need to determine the electromotive force (emf) induced at different positions of the coil: inside, partly inside, and outside the field.

Step 2: Calculating the Induced EMF Inside the Field (Part a)

When the coil is completely inside the magnetic field, there is no change in the area of the coil exposed to the field over time. According to Faraday's law of induction, the induced emf is given by \[ \text{emf} = -\frac{d\Phi_B}{dt} = -B \frac{dA}{dt} \]where \(\Phi_B\) is the magnetic flux and \(A\) is the area. Since \(\frac{dA}{dt} = 0\) for a coil fully inside the magnetic field, the induced emf is 0 V.

Step 3: Calculating the Induced EMF Partly Inside the Field (Part b)

When the coil is partly inside the field, the change in flux is due to the change in the area of the coil outside the field. The induced emf is given by\[ \text{emf} = -B \frac{dA}{dt} \]The area \(A(t)\) being removed per second is \(\text{width} \times \text{velocity} = 30.0\, \text{cm} \times 2.00\, \text{cm/s} = 60.0\, \text{cm}^2/s\). Therefore, the emf is \[ \text{emf} = -1.25\, \text{T} \times 60.0\, \text{cm}^2/s = -75.0\, \text{mV}. \]

Step 4: Calculating the Induced EMF Outside the Field (Part c)

When the coil is completely outside the magnetic field, there is no magnetic flux through the coil, and thus there is no change in magnetic flux over time. Consequently, as per Faraday's law,\[ \text{emf} = 0 \text{ V} \].

Key Concepts

Magnetic Flux

Magnetic flux is a fundamental concept in electromagnetism that helps us understand how magnetic fields interact with different areas. It is represented as \( \Phi_B \), where \( B \) is the magnetic field strength and \( A \) is the area through which the field lines pass.
The unit for magnetic flux is the Weber (Wb). Magnetic flux is crucial because any change in it can induce electromotive force (emf), as described by Faraday’s Law of Induction.

Magnetic flux can be thought of as the "amount of magnetic field" passing through a given area. It is determined by three factors:

• The strength of the magnetic field (B)
• The area (A) observed in the context of the flux
• The angle between the magnetic field lines and the surface area normal, described by the formula: \[ \Phi_B = B \times A \times \cos(\theta) \].

In the given exercise, magnetic flux is significant because, when the coil moves out of the magnetic field, changes in flux result in emf. When the coil is wholly inside or outside the field, there is no change in flux and the induced emf is zero.

Electromotive Force (EMF)

Electromotive force (emf) is a vital concept in electrical circuits and is closely related to magnetic flux changes. It is the potential difference generated by a change in magnetic flux and can be thought of as the "driving force" that pushes charges around a circuit. In simple terms, emf is like a pressure difference that causes electric current to flow.
According to Faraday's Law, the induced emf is:\[ \text{emf} = -\frac{d\Phi_B}{dt} \] which means the emf is proportional to the rate of change of magnetic flux over time.

There are some key points about emf:

• It is generated without any external power source, solely due to the movement of the coil or a changing magnetic field.
• The negative sign in Faraday’s law (Lenz’s Law) indicates that the induced emf creates a magnetic field opposing the change in the original magnetic flux.
• When a coil is moving completely inside a uniform magnetic field, there is no change in flux and hence, no emf.

This principle is used practically in the given exercise; when the coil begins moving out of the magnetic field, creating a change in flux, emf is induced.

Uniform Magnetic Field

A uniform magnetic field is characterized by having magnetic field lines that are evenly spaced and parallel, indicating the same magnitude and direction throughout the region. This consistency is crucial in simplifying the calculation of phenomena such as magnetic flux and, consequently, the induced emf.
Understanding uniform magnetic fields is essential because it allows for easy application of Faraday’s Law, as the magnetic field strength remains constant over the entire area of interest.

Here are some useful points about uniform magnetic fields:

• The field strength (\( B \)) does not change, making calculations straightforward for areas fully within the field.
• Changes in the position of an object within such a field can lead to changes in magnetic flux if the area exposed to the field changes.
• In the exercise, the uniform magnetic field simplifies the determination of when changes in area (as the coil moves) lead to the induction of emf. Specifically, when the coil is partly in and partly out of the uniform magnetic field.

By comprehending uniform magnetic fields, students can better predict and calculate changes in systems where magnetic induction occurs.