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Chapter 29: Problem 63

A slender rod, 0.240 \(\mathrm{m}\) long, reates with an angular speed of 8.80 \(\mathrm{rad} / \mathrm{s}\) about an axis through one end and perpendicular to the rod. The plane of rotation of the rod is perpendicular to a uniform magnetic field with a magnitude of 0.650 \(\mathrm{T}\) (a) What is the induced emf in the rod? (b) What is the potential difference between its ends? (c) Suppose instead the rod rotates at 8.80 \(\mathrm{rad} / \mathrm{s}\) about an axis through its center and perpendicular to the rod, In this case, what is the potential difference between the ends of the rod? Between the center of the rod and one end?

Short Answer

Expert verified
(a) 0.164 V; (b) 0.164 V; (c) 0.082 V; 0.041 V for center to one end.

Step by step solution

01

Understanding the problem

We need to calculate the induced emf in the rotating rod and the potential difference between its ends for two different cases. In the first scenario, the rod rotates about an axis through one end. In the second, it rotates about its center.
02

Calculating emf for the rod rotating about one end

The formula for the induced emf () in a rod rotating about one end is given by = \(\frac{1}{2} B \omega L^2\) where \( B \) is the magnetic field, \( \omega \) is the angular speed, and \( L \) is the length of the rod.Substitute the values:\[ \begin{align*} \varepsilon & = \frac{1}{2} \times 0.650 \times 8.80 \times (0.240)^2\ & = \frac{1}{2} \times 0.650 \times 8.80 \times 0.0576\ & \approx 0.164 V. \end{align*} \]
03

Potential difference for the rod rotating about one end

The potential difference between the ends of the rod when it rotates about one end equals the induced emf. Therefore, \( V = 0.164 \, V \).
04

Calculating potential difference for the rod rotating about its center

When the rod rotates about its center, the induced emf is calculated slightly differently; it creates two emf values on each half of the rod. The potential difference for one half of the rod is given by:\[ \varepsilon_{\text{half}} = \frac{1}{2} B \omega (\frac{L}{2})^2 \approx \frac{1}{2} \times 0.650 \times 8.80 \times \left(\frac{0.240}{2}\right)^2 \]Calculate:\[ \varepsilon_{\text{half}} = \frac{1}{2} \times 0.650 \times 8.80 \times 0.0144 = 0.041 V \]The potential difference between the ends of the rod is twice this value: \[ V = 2 \times 0.041 = 0.082 \, V \].
05

Potential difference between the center and one end

Since the entire difference is partitioned equally across both halves of the rod when rotating about the center, the potential difference between the center and one end is also given by one half of the full potential difference:\( V = 0.041 \, V \).

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Key Concepts

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

Magnetic Field Interaction
Whenever a magnetic field interacts with a moving conductor, such as a rotating rod, an electric potential is induced. This is a direct consequence of Faraday's Law of Induction, which states that a change in magnetic flux through a circuit induces an electromotive force (EMF). In this case, the rotating rod is the moving part in the magnetic field, causing this change in flux.

The magnetic field, denoted by \(B\), is uniform and acts perpendicular to the plane of rotation, enhancing the induction effect. The strength of the induced EMF can be quantified using the formula \(\varepsilon = \frac{1}{2}B\omega L^2\). Here, \(\omega\) represents angular velocity and \(L\) is the length of the rod.

This setup exemplifies how moving conductors in magnetic fields can be utilized to create electric currents, which is the principle behind many power generation technologies.
Rotational Motion
Rotational motion plays a crucial role in this problem. The rod's angular velocity, \(\omega\), at 8.80 \(\text{rad/s}\), signifies how fast the rod spins around a fixed axis. When discussing rotation about one end of the rod, we consider how the entire length of the rod sweeps through space.

The rotational motion about one end results in one side of the rod moving faster than the center, causing a large change in speed across its length. This results in a higher induced EMF compared to rotating about the center. When the rod rotates about its center, both halves move at identical speeds.

Understanding how EMF is induced in rotating systems provides an insight into mechanical systems that transform rotational movement into electricity. The position of the axis in relation to the center of mass can significantly affect the potential difference created.
Physics Problem Solving
To solve physics problems such as this one, it's important to systematically break down the tasks using known formulas and careful observations.

Start by identifying key variables and the scenario details: the magnetic field (\(B\), given as \(0.650\ \text{T}\)), the rod's length (\(L\)), and its angular speed (\(\omega\)). Analyzing these elements helps in applying appropriate physics laws, such as Faraday鈥檚 Law, to derive solutions.

The problem requires considering two rotational scenarios鈥攁bout one end and about the center. Each case uses specific formulas for EMF and potential difference, reflecting the physical arrangement's effect on the outcome.
  • For rotation about one end, use \(\varepsilon = \frac{1}{2} B \omega L^2 \).
  • For the center, calculate EMF for half and double to get total \(V\).
Practicing such problems enhances understanding of electromagnetic induction, providing tools for tackling more complex systems.

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Most popular questions from this chapter

A closely wound search coil (Exercise 29.3) has an area of \(3.20 \mathrm{cm}^{2}, 120\) turns, and a resistance of \(60.0 \Omega .\) It is connected to a charge-measuring instrument whose resistance is 45.0\(\Omega\) . When the coil is rotated quickly from a position parallel to a uniform magnetic field to a position perpendicular to the field, the instrument indicates a charge of \(3.56 \times 10^{-5} \mathrm{C}\) . What is the magnitude of the field?

Antenna emf. A satellite, orbiting the earth at the equator at an altitude of 400 \(\mathrm{km}\) , has an antenna that can be modeled as a \(2.0-\mathrm{m}-\) long rod. The antenna is oriented perpendicular to the earth's surface. At the cquator, the earth's magnetic field is cssentially horizontal and has a value of \(8.0 \times 10^{-5} \mathrm{T}\) ; ignore any changes in \(B\) with altitude. Assuming the orbit is circular, determine the induced emf between the tips of the antenna.

emf in a Bullet. At the equator, the earth's magnetic field is approximately horizontal, is directed towand the north, and has a value of \(8 \times 10^{-5} \mathrm{T}\) . (a) Estimate the emf induced between the top and bottom of a bullet shot horizontally at a target on the equator if the bullet is shot toward the east. Assume the bullet has a length of 1 \(\mathrm{cm}\) and a diamcter of 0.4 \(\mathrm{cm}\) and is traveling at 300 \(\mathrm{m} / \mathrm{s}\) . Which is at higher potential: the top or boutom of the bullet? (b) What is the emfif the bullet travels south?(c) What is the emf induced between the front and back of the bullet for any horizontal velocity?

The armature of a small generator consists of a flat, square coil with 120 turns and sides with a length of 1.60 \(\mathrm{cm}\) . The coil rotates in a magnetic fleld of 0.0750 T. What is the angular speed of the coil if the maximum emf produced is 24.0 \(\mathrm{mV} ?\)

A capacitor has two parallel plates with area \(A\) separated by a distance \(d\) . The space between plates is filled with a material having dielectric constant \(K\) . The material is not a perfect insulator but has resistivity \(\rho\) . The capacitor is initially charged with charge of magnitude \(Q_{0}\) on each plate that gradually discharges by conduction through the dielectric. (a) Calculate the conduction current density \(j_{\mathrm{C}}(t)\) in the dielectric. (b) Show that at any instant the dis-placement current density in the diclectric is equal in magnitude to the oonduotion current density but opposite in direction, so the total current density is zero at every instant.
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