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Chapter 29: Problem 67

CALC A slender rod, 0.240 m long, rotates 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) 0.1647 V; (b) 0.1647 V; (c) Ends: 0.0824 V, Center to end: 0.0412 V.

Step by step solution

01

Calculate the induced emf for end rotation

The formula for the induced emf (electromotive force) in a rotating rod of length \(l\) about an axis perpendicular to a magnetic field \(B\) is given by \( \epsilon = \frac{1}{2} B \omega l^2 \), where \( \omega \) is the angular speed. Substituting the given values, \( B = 0.650 \, T \), \( \omega = 8.80 \, rad/s \), and \( l = 0.240 \, m \), we have:\[ \epsilon = \frac{1}{2} \times 0.650 \times 8.80 \times (0.240)^2 \]Calculating, \( \epsilon = 0.164736 \, V \).
02

Calculate the potential difference for end rotation

When the rod is rotating about one end, the induced emf itself across the length of the rod is the potential difference between its ends. Therefore, the potential difference is also \( 0.164736 \, V \).
03

Calculate the potential difference for center rotation

Now, if the rod rotates about its center, the induced emf is different. The emf's formula in this configuration is \( \epsilon = \frac{1}{4} B \omega l^2 \) because only half the rod length contributes to the emf when rotating about the center. Substituting the values:\[ \epsilon = \frac{1}{4} \times 0.650 \times 8.80 \times (0.240)^2 \]Calculating, \( \epsilon = 0.082368 \, V \). This is the potential difference between the ends.
04

Calculate the potential difference between center and one end for center rotation

Since the emf across the whole rod when rotating about the center is \( 0.082368 \, V \), the potential difference between the center and one end would be half this value because the rod is symmetric about the center. Thus, the potential difference is:\[ \frac{0.082368}{2} = 0.041184 \, V \].

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Key Concepts

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

Induced EMF
In electromagnetic induction, induced electromotive force (EMF) occurs when a conductor, such as a rod, moves through a magnetic field or when the magnetic field around the conductor changes. The phenomenon takes advantage of Faraday's Law of Induction, which describes how an electric current is generated in a circuit due to a changing magnetic field.
The induced EMF (\(\epsilon\)) is a crucial concept because it demonstrates the conversion of mechanical energy into electrical energy. This is why EMF is often described as an energy-producing force in electrical systems.
  • The formula for calculating the induced EMF in a rotating rod of length \(l\) and angular speed \(\omega\) in a magnetic field \(B\) perpendicular to the rotation is \( \epsilon = \frac{1}{2} B \omega l^2 \).
  • Substituting the values of the problem (\(B = 0.650 \, T\), \(\omega = 8.80 \, rad/s\), and \(l = 0.240 \, m\)) into this formula allows us to find the induced EMF of 0.164736 V.
This formula is derived from integration as the rod sweeps through the magnetic field, where every elemental part of the rod contributes to the total EMF exerted across its length.
Magnetic Field
A magnetic field is an invisible force that surrounds magnetic materials and influences the environment. This field is created by either permanent magnets or electric currents, and it's essential for understanding electromagnetic induction.
When a conductor such as a rod moves through a magnetic field, the field exerts a force on the charges within the conductor, leading to a movement of electrons and thus an induced current.
  • The strength of this magnetic field is denoted by \(B\) and measured in Teslas (T).
  • In our exercise, the magnetic field strength is 0.650 T, which is a measure of how intensely the field exerts force on the rod.
The direction and speed at which a magnetic field acts are critical factors in determining the magnitude of the induced EMF, as depicted by the perpendicular relationship between the field and the plane of rotation of the rod in the exercise.
Rotational Motion
Rotational motion refers to the movement of an object around a central axis. This type of motion is commonly encountered in scenarios involving wheels, fans, and in our exercise鈥攁 rotating rod.
Angular speed \(\omega\), which measures how quickly an object is rotating, is integral to calculating the induced EMF. It is measured in radians per second (rad/s), providing a sense of the rotational velocity.
  • For the rod in our exercise, angular speed is \(8.80 \, rad/s\), contributing to the dynamic interaction of the rod with the magnetic field.
  • When motion occurs about one end of the rod, the entire length contributes to the induced EMF calculation.
  • Conversely, when rotating about the center, only half the length directly influences the EMF, as each half moves in opposite directions relative to the center.
In essence, understanding rotational motion helps us ascertain how various rotation points alter the electric potential distribution in a magnetic field, which is critical for applications in devices like electric generators.

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Most popular questions from this chapter

The magnetic field within a long, straight solenoid with a circular cross section and radius \(R\) is increasing at a rate of \(d B / d t\) . (a) What is the rate of change of flux through a circle with radius \(r_{1}\) inside the solenoid, normal to the axis of the solenoid, and with center on the solenoid axis? (b) Find the magnitude of the induced electric field inside the solenoid, at a distance \(r_{1}\) from its axis. Show the direction of this field in a diagram. (c) What is the magnitude of the induced electric field outside the solenoid, at a distance \(r_{2}\).from the axis? (d) Graph the magnitude of the induced electric field as a function of the distance \(r\) from the axis from \(r=0\) to \(r=2 R\) (e) What is the magnitude of the induced emf in a circular turn of radius \(R / 2\) that has its center on the solenoid axis? (f) What is the magnitude of the induced emf if the radius in part (e) is \(R ?\) (g) What is the induced emf if the radius in part (e) is 2R?

A circular loop of wire is in a region of spatially uniform magnetic field, as shown in Fig. E29.15. The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) \(B\) is increasing; (b) \(B\) is decreasing; (c) \(B\) is constant with value \(B_{0} .\) Explain your reasoning.

At temperatures near absolute zero, \(B_{\mathrm{c}}\) approaches 0.142 \(\mathrm{T}\) for vanadium, a type-I superconductor. The normal phase of vanadium has a magnetic susceptibility close to zero. Consider a long, thin vanadium cylinder with its axis parallel to an external magnetic field \(\vec{B}_{0}\) in the \(+x\) -direction. At points far from the ends of the cylinder, by symmetry, all the magnetic vectors are parallel to the \(x\) -axis. At temperatures near absolute zero, what are the resultant magnetic field \(\vec{B}\) and the magnetization \(\vec{M}\) inside and outside the cylinder (far from the ends ) for (a) \(\vec{B}_{0}=(0.130 \mathrm{T}) \hat{\imath}\) and (b) \(\vec{\boldsymbol{B}}_{0}=(0.260 \mathrm{T}) \hat{\mathfrak{t}} ?\)

CALC In Fig. 29.22 the capacitor plates have area 5.00 \(\mathrm{cm}^{2}\) and separation 2.00 \(\mathrm{mm}\) . The plates are in vacuum. The charging current \(i_{\mathrm{C}}\) has a constant value of 1.80 \(\mathrm{mA} .\) At \(t=0\) the charge on the plates is zero. (a) Calculate the charge on the plates, the electric field between the plates, and the potential difference between the plates when \(t=0.500 \mu \mathrm{s}\) (b) Calculate \(d E / d t\) , the time rate of change of the electric field between the plates. Does \(d E / d t\) vary in time? (c) Calculate the displacement current density \(j_{\mathrm{D}}\) between the plates, and from this the total displacement current \(i_{\mathrm{D}} .\) How do \(i_{\mathrm{C}}\) and \(i_{\mathrm{D}}\) compare?

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. E29.24). 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.
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