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Chapter 13: Problem 14

Show that \(\varepsilon\) and \(d \Phi_{\mathrm{m}} / d t\) have the same units.

Short Answer

Expert verified
In this exercise, we showed that the units of \(\varepsilon\) (electromotive force) and \(d \Phi_{\mathrm{m}}/dt\) (rate of change of magnetic flux) are the same. We found that \(\varepsilon\) has units of V (volts) or J/C (joules per coulomb), while \(d \Phi_{\mathrm{m}}/dt\) has units of T路m虏/s (tesla meter squared per second). Using Faraday's law of electromagnetic induction, we related the units of these quantities, showing that V = J/C = T路m虏/s. Hence, we proved that their units are indeed the same.

Step by step solution

01

Find the units of \(\varepsilon\)

The electromotive force \(\varepsilon\) is a measure of the electrical potential created by a source, such as a battery or generator. The units of \(\varepsilon\) are volts (V), which can also be expressed as joules per coulomb (J/C), as voltage represents the energy required to move a unit of charge across an electrical potential.
02

Find the units of \(d \Phi_{\mathrm{m}}\)

The magnetic flux \(\Phi_{\mathrm{m}}\) represents the amount of magnetic field passing through a given area, and is calculated as the product of the magnetic field strength and the area through which it passes. Magnetic flux is measured in units of webers (Wb). A weber can be expressed as a tesla meter squared (T路m虏), as the magnetic field strength is measured in teslas (T) and the area is measured in square meters (m虏).
03

Find the units of \(dt\)

The \(dt\) term represents a small change in time. Time is typically measured in seconds (s).
04

Find the units of \(d \Phi_{\mathrm{m}}/dt\)

Now, we need to find the units of the derivative \(d \Phi_{\mathrm{m}}/dt\). This represents the rate of change of the magnetic flux with respect to time. To find the units of this quantity, we divide the units of \(d \Phi_{\mathrm{m}}\) by the units of \(dt\): \[\frac{\text{T路m虏}}{\text{s}}\]
05

Relate the units of \(\varepsilon\) and \(d \Phi_{\mathrm{m}}/dt\)

We know that \(\varepsilon\) has units of V or J/C, and \(d \Phi_{\mathrm{m}}/dt\) has units of T路m虏/s. To show that they are the same, we need to relate the units of \(\varepsilon\) to those of \(d \Phi_{\mathrm{m}}/dt\) through a known relationship. According to Faraday's law of electromagnetic induction, the emf induced in a circuit is proportional to the rate of change of magnetic flux through the circuit. Mathematically, this is expressed as: \[\varepsilon = - \frac{d \Phi_{\mathrm{m}}}{dt}\] Using the units previously determined for \(\varepsilon\) and \(d \Phi_{\mathrm{m}}/dt\), we see that: \[\text{V} = \text{J/C} = \frac{\text{T路m虏}}{\text{s}}\] Since the units of \(\varepsilon\) and \(d \Phi_{\mathrm{m}}/dt\) are the same, we have shown that \(\varepsilon\) and \(d \Phi_{\mathrm{m}}/dt\) have the same units.

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Key Concepts

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

Faraday's Law of Electromagnetic Induction
Faraday's Law of Electromagnetic Induction is a fundamental principle in physics that describes how electric currents can be generated through changing magnetic fields. It tells us that an electromotive force (emf) is induced in a circuit whenever the magnetic flux through the circuit changes.
This principle is a foundation for many electrical devices, such as transformers and electric generators, which rely on converting mechanical energy into electrical energy and vice versa.
  • An emf is generated in a closed loop when there is a change in magnetic flux.

  • The induced emf is directly proportional to the rate of change of magnetic flux.

  • The direction of the induced emf always opposes the change in flux, as per Lenz's law.
Faraday's law is mathematically expressed as:\[\varepsilon = -\frac{d \Phi_{\mathrm{m}}}{d t}\]This equation shows that the emf \( \varepsilon \) is equal to the negative rate of change of magnetic flux \( d\Phi_{\mathrm{m}}/dt \), ensuring that these two quantities have matching units.
Magnetic Flux
Magnetic flux is a measure of the magnetic field passing through a certain area. It gives us an understanding of the strength and distribution of magnetic fields over a surface.
  • Magnetic flux is denoted by \( \Phi_{\mathrm{m}} \).

  • It is calculated as the product of the magnetic field strength and the area it penetrates.

  • It is often measured when discussing the effects of magnetic fields in various processes and devices.
The unit of measurement for magnetic flux is the weber (Wb), where a single weber is equivalent to one tesla meter squared \( \text{T} \cdot \text{m}^2 \). Magnetic flux plays a crucial role in Faraday's Law of Electromagnetic Induction, as any change in flux will induce an emf.
Electric Potential
Electric potential is the work required to move a unit charge from a reference point to a specific point inside an electric field, without producing any acceleration. In simpler terms, it is a measure of the potential energy per unit charge at a given point.
  • Electric potential is commonly referred to as voltage.

  • It reflects the potential energy per charge available for performing work.

  • It is crucial in understanding circuits, batteries, and electrical fields.
The unit for electric potential is the volt (V), which can also be expressed in terms of energy per charge as joules per coulomb (J/C). Electric potential is an essential concept in electrostatics and provides a measure for the energy landscape that charges experience.
Units of Measurement
Units of measurement are fundamental in science as they provide a standardized way to express physical quantities.
  • They allow scientists and engineers to communicate findings and understand the magnitude of different quantities.

  • Units are critical for verifying equations and physical laws, like Faraday's law.

  • Common units in electromagnetism include volts, webers, and seconds.
In the context of Faraday's Law, the equivalence of the units for electromotive force (\( \varepsilon \)) and the rate of change of magnetic flux (\( d\Phi_{\mathrm{m}}/dt \)) is important for validating physical relationships. For instance:
  • \( \varepsilon \) unit: Volt (V) or Joule per coulomb (J/C).

  • \( d \Phi_{\mathrm{m}}/dt \) unit: Weber per second (Wb/s) or Tesla meter squared per second (T路m虏/s).
Ensuring that units match across equations helps confirm that the physical interpretation and calculations are correct across different scenarios.

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Most popular questions from this chapter

A long solenoid of radius \(a\) with \(n\) turns per unit length is carrying a time-dependent current \(I(t)=I_{0} \sin \omega t \quad\) where \(\quad I_{0}\) and \(\omega\) are constants. The solenoid is surrounded by a wire of resistance \(R\) that has two circular loops of radius \(b\) with \(b>a\). Find the magnitude and direction of current induced in the outer loops at time \(t=0\).

The normal to the plane of a single-turn conducting loop is directed at an angle \(\theta\) to a spatially uniform magnetic field \(\overrightarrow{\mathbf{B}}\). It has a fixed area and orientation relative to the magnetic field. Show that the emf induced in the loop is given by \(\varepsilon=(d B / d t)(A \cos \theta), \) where \(A\) is the area of the loop.

A 50-turn rectangular coil with dimensions \(0.15 \mathrm{m} \times 0.40 \mathrm{m}\) rotates in a uniform magnetic field of magnitude \(0.75 \mathrm{T}\) at 3600 rev/min. (a) Determine the emf induced in the coil as a function of time. (b) If the coil is connected to a \(1000-\Omega\) resistor, what is the power as a function of time required to keep the coil turning at 3600 rpm? (c) Answer part (b) if the coil is connected to a 2000 \(\Omega\) resistor.

When a magnetic field is first turned on, the flux through a 20 -turn loop varies with time according to \(\Phi_{\mathrm{m}}=5.0 t^{2}-2.0 t,\) where \(\Phi_{\mathrm{m}}\) is in milliwebers, \(t\) is in seconds, and the loop is in the plane of the page with the unit normal pointing outward. (a) What is the emf induced in the loop as a function of time? What is the direction of the induced current at (b) \(t=0,(\mathrm{c}) 0.10,\) (d) \(1.0,\) and (e) 2.0 s?

A circular loop of wire of radius \(10 \mathrm{cm}\) is mounted on a vertical shaft and rotated at a frequency of 5 cycles per second in a region of uniform magnetic field of 2 Gauss perpendicular to the axis of rotation. (a) Find an expression for the time-dependent flux through the ring. (b) Determine the time-dependent current through the ring if it has a resistance of \(10 \Omega\).
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