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Chapter 29: Problem 5

A circular loop of wire with a radius of 12.0 cm and oriented in the horizontal \(xy\)-plane is located in a region of uniform magnetic field. A field of 1.5 T is directed along the positive z-direction, which is upward. (a) If the loop is removed from the field region in a time interval of 2.0 ms, find the average emf that will be induced in the wire loop during the extraction process. (b) If the coil is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

Short Answer

Expert verified
(a) The average induced emf is approximately 106.5 V. (b) The induced current is counterclockwise when viewed from above.

Step by step solution

01

Understand Faraday's Law of Induction

Faraday's Law states that the average electromotive force (emf) induced in a loop is equal to the rate of change of the magnetic flux through the loop. It can be expressed as: \[ \text{emf} = - \frac{\Delta \Phi}{\Delta t} \]where \( \Delta \Phi \) is the change in magnetic flux, and \( \Delta t \) is the time interval.
02

Calculate Initial Magnetic Flux

The magnetic flux \( \Phi \) through the loop is given by:\[ \Phi = B \cdot A = B \cdot \pi r^2 \]where \( B \) is the magnetic field, and \( A \) is the area of the loop. For a radius \( r = 0.12 \) m and \( B = 1.5 \) T, we compute:\[ \Phi = 1.5 \times \pi \times (0.12)^2 \]
03

Perform the Flux Calculation

Now, actually calculate the initial magnetic flux:\[ \Phi = 1.5 \times \pi \times 0.0144 = 0.0679 \pi \]This evaluates approximately to \( \Phi \approx 0.213 \) Wb.
04

Determine Change in Flux

When the loop is removed from the magnetic field, the final magnetic flux is zero. Therefore, the change in flux is:\[ \Delta \Phi = 0 - 0.213 = -0.213 \text{ Wb} \]
05

Calculate the Average Induced EMF

Using Faraday's Law:\[ \text{emf} = -\frac{\Delta \Phi}{\Delta t} = -\frac{-0.213}{0.002} \]\[ \text{emf} \approx 106.5 \text{ V} \]
06

Determine Direction of Induced Current

According to Lenz's Law, the direction of the induced current is to oppose the change in magnetic flux. Since the wire loop was initially in the field and the field is removed, the induced current will have a magnetic moment that opposes the loss of upward field. Thus, when viewed from above, the current is induced in a direction that opposes the field's decrease - which is counterclockwise.

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Key Concepts

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

Electromagnetic Induction
Electromagnetic induction is a fundamental principle of physics describing how an electric current is generated by changing magnetic fields. It was first discovered by Michael Faraday. This process is the foundation of many technological applications, such as electric generators and transformers.

The key idea is that a moving or changing magnetic field can produce an electric current in a conductor. This action relies on the relative motion between the magnetic field and the conductor, thus creating an electromotive force (EMF).

It is important to understand that it's not the strength of the magnetic field alone that causes induction, but how it changes over time.
Magnetic Flux
Magnetic flux (\( \Phi \)) is a measure of the number of magnetic field lines passing through a given area. You can think of it as the total magnetic field "threading" through a particular space. When dealing with a loop of wire, the magnetic flux is determined using the formula:\[ \Phi = B \times A \]where \( B \) is the magnetic field strength and \( A \) is the area of the loop.

In the exercise, the magnetic flux was calculated as a product of the magnetic field and the area of the circular wire loop. The more lines of magnetic field that pass through the loop, the greater the magnetic flux.
Lenz's Law
Lenz's Law gives direction to the induced current and EMF. It states that an induced current will flow in a direction that opposes the change that caused it. This concept can be a bit tricky at first, but it is a natural consequence of the law of conservation of energy.

In the given exercise, as the magnetic flux decreases to zero when the loop is removed from the magnetic field, an induced current circulates in the loop to maintain the original flux direction. This opposing action ensures that energy is conserved in the system.
Induced EMF
The induced electromotive force (EMF) is essentially the voltage generated by the process of electromagnetic induction. According to Faraday's law, the magnitude of the induced EMF in a closed loop is equal to the rate of change of magnetic flux through the loop:\[ \text{emf} = - \frac{\Delta \Phi}{\Delta t} \]

The negative sign indicates the direction of the induced EMF, as explained by Lenz's Law. In simple terms, faster changes in magnetic flux induce a stronger EMF.
Circular Wire Loop
A circular wire loop is a simple yet effective setup for observing electromagnetic induction. The symmetry of the circle allows for straightforward calculations of area, making it easier to apply equations related to flux and EMF.

In our problem, a circular loop with a given radius was used to determine the magnetic flux. The circular configuration ensures that calculations related to changes in the magnetic field across the loop are uniform and the resulting induced EMF can be accurately quantified.
Magnetic Field
A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. In this exercise, the magnetic field is uniformly directed along the z-axis at 1.5 T. The uniformity and orientation of the field are critical as they ensure consistent induction across the loop.

Magnetic fields are represented by field lines. The direction of these lines goes from the north to the south pole outside the magnet and vice versa inside it.

The uniform magnetic field in the exercise plays a crucial role as it builds a foundation for calculating magnetic flux through the loop.
Direction of Current
The direction of the induced current in a loop can be determined using Lenz's Law. In this context, we had a loop that was initially experiencing a vertical magnetic field which was then removed. The induced current resisted this change, trying to "hold onto" the magnetic field.

Therefore, when viewed from above, the induced current flows counterclockwise, creating a field in the opposite direction to the change 鈥� attempting to maintain the original upward field.
  • This concept is crucial for understanding how electric generators work, where the direction of rotation is key to harnessing current effectively.

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Most popular questions from this chapter

A single loop of wire with an area of 0.0900 m\(^2\) is in a uniform magnetic field that has an initial value of 3.80 T, is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 T/s. (a) What emf is induced in this loop? (b) If the loop has a resistance of 0.600 \(\Omega\), find the current induced in the loop.

You are shipwrecked on a deserted tropical island. You have some electrical devices that you could operate using a generator but you have no magnets. The earth's magnetic field at your location is horizontal and has magnitude 8.0 \(\times\) 10\(^{-5}\) T, and you decide to try to use this field for a generator by rotating a large circular coil of wire at a high rate. You need to produce a peak emf of 9.0 V and estimate that you can rotate the coil at 30 rpm by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum number of turns the coil can have is 2000. (a) What area must the coil have? (b) If the coil is circular, what is the maximum translational speed of a point on the coil as it rotates? Do you think this device is feasible? Explain.

A very long, rectangular loop of wire can slide without friction on a horizontal surface. Initially the loop has part of its area in a region of uniform magnetic field that has magnitude \(B =\) 2.90 T and is perpendicular to the plane of the loop. The loop has dimensions 4.00 cm by 60.0 cm, mass 24.0 g, and resistance \(R =\) 5.00 \(\times\) 10\(^{-3} \Omega\). The loop is initially at rest; then a constant force \(F_{ext}\) = 0.180 N is applied to the loop to pull it out of the field (Fig. P29.46). (a) What is the acceleration of the loop when \(v =\) 3.00 cm/s? (b) What are the loop's terminal speed and acceleration when the loop is moving at that terminal speed? (c) What is the acceleration of the loop when it is completely out of the magnetic field?

A long, straight solenoid with a cross-sectional area of 8.00 cm\(^2\) is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 A. A second winding of 12 turns encircles the solenoid at its center. The current in the solenoid is turned off such that the magnetic field of the solenoid becomes zero in 0.0400 s. What is the average induced emf in the second winding?

A circular conducting ring with radius \(r_0 =\) 0.0420 m lies in the xy-plane in a region of uniform magnetic field \(\overrightarrow{B} = B_0 [1 - 3(t/t_0)^2 + 2(t/t_0)^3]\hat{k}\). In this expression, \(t_0 =\) 0.0100 s and is constant, \(t\) is time, \(\hat{k}\) is the unit vector in the +\(z\)-direction, and \(B_0\) = 0.0800 T and is constant. At points \(a\) and \(b\) (Fig. P29.58) there is a small gap in the ring with wires leading to an external circuit of resistance \(R =\) 12.0 \(\Omega\). There is no magnetic field at the location of the external circuit. (a) Derive an expression, as a function of time, for the total magnetic flux \(\Phi_B\) through the ring. (b) Determine the emf induced in the ring at time \(t =\) 5.00 \(\times\) 10\(^{-3}\) s. What is the polarity of the emf? (c) Because of the internal resistance of the ring, the current through \(R\) at the time given in part (b) is only 3.00 mA. Determine the internal resistance of the ring. (d) Determine the emf in the ring at a time \(t =\) 1.21 \(\times\) 10\(^{-2}\) s. What is the polarity of the emf? (e) Determine the time at which the current through \(R\) reverses its direction.
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