Physics
Induced Electric Field Formula
The induced electric field formula describes the electric field generated by a changing magnetic field. It is given by the equation E = -dΦ/dt, where E is the induced electric field, Φ is the magnetic flux, and t is time. This formula is a fundamental concept in electromagnetism and is used to understand electromagnetic induction and Faraday's law.
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10 Key excerpts on "Induced Electric Field Formula"
eBook - PDF- Daniel Fleisch(Author)
- 2008(Publication Date)
- Cambridge University Press(Publisher)
As always, you should remember that these fields exist in three dimensions, and you can see full 3-D visualizations on the book’s website. A student’s guide to Maxwell’s Equations 62 Examples of a charge-based and an induced electric field are shown in Figure 3.1 . Note that the induced electric field in Figure 3.1(b) is directed so as to drive an electric current that produces magnetic flux that opposes the change in flux due to the changing magnetic field. In this case, the motion of the magnet to the right means that the leftward magnetic flux is decreasing, so the induced current produces additional leftward magnetic flux. Here are a few rules of thumb that will help you visualize and sketch the electric fields produced by changing magnetic fields: Induced electric field lines produced by changing magnetic fields must form complete loops. The net electric field at any point is the vector sum of all electric fields present at that point. Electric field lines can never cross, since that would indicate that the field points in two different directions at the same location. In summary, the ~ E in Faraday’s law represents the induced electric field at each point along path C , a boundary of the surface through which the magnetic flux is changing over time. The path may be through empty space or through a physical material – the induced electric field exists in either case. Faraday’s law 63 H C ðÞ dl The line integral To understand Faraday’s law, it is essential that you comprehend the meaning of the line integral. This type of integral is common in physics and engineering, and you have probably come across it before, perhaps when confronted with a problem such as this: find the total mass of a wire for which the density varies along its length. This problem serves as a good review of line integrals. Consider the variable-density wire shown in Figure 3.2(a) .
eBook - PDF- Bhag Singh Guru, Hüseyin R. Hiziroglu(Authors)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
This equation enables us to compute the electric field intensity at a fixed point in space when the magnetic field is a function of time. Note that for a static field, it reduces to ∇ × E = 0. We also identify (7.15) as Maxwell’s equation (Faraday’s law) in the integral form . We can use this equation to calculate the induced emf around a stationary closed path. As mentioned in Section 7.3, this equation is then the integral form of the transformer equation. We can also express it as e t = − s ∂ B ∂ t · d s (7.17) where the subscript t is added to indicate that it is the transformer emf only. 7.4.1 General equations The motion of a closed loop (circuit) in a magnetic field produces a motional emf in that loop as stated by (7.6). If we use the subscript m 288 7 Time-varying electromagnetic fields to indicate the motional emf, we can write (7.6) for the closed circuit as e m = c ( u × B ) · d (7.18) When the loop is moving in a time-varying magnetic field, the total induced emf will be e = e t + e m = − s ∂ B ∂ t · d s + c ( u × B ) · d (7.19) The direction of contour c in this equation defines the direction of the unit normal to the surface ds in accordance with the right-hand rule. This equation is another general statement of Faraday’s law of induc-tion . In terms of the induced electric field, equation (7.19) can also be written as c E · d = − s ∂ B ∂ t d s + c ( u × B ) · d The application of Stokes’ theorem yields s ( ∇ × E ) · d s = − s ∂ B ∂ t d s + x [ ∇ × ( u × B )] · d s Since the surface s is bounded by an arbitrary contour c , and for the equation to be valid in general, we can equate the integrands and obtain ∇ × E = − ∂ B ∂ t + ∇ × ( u × B ) (7.20) This equation is the most general form of Maxwell’s equation (Faraday’s law) in the point form . It enables us to determine the electric field at a point of observation moving with a velocity u in a magnetic field B .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- University Publications(Publisher)
This case is called an induced EMF. On the other hand, when the magnet is stationary and the conductor is rotated, the moving charges experience a magnetic force (as described by the Lorentz force law), and this magnetic force pushes the charges through the wire. This case is called motional EMF. Electrical motor An electrical generator can be run backwards to become a motor. For example, with the Faraday disc, suppose a DC current is driven through the conducting radial arm by a voltage. Then by the Lorentz force law, this traveling charge experiences a force in the magnetic field B that will turn the disc in a direction given by Fleming's left hand rule. In the absence of irreversible effects, like friction or Joule heating, the disc turns at the rate necessary to make d Φ B / dt equal to the voltage driving the current. ____________________ WORLD TECHNOLOGIES ____________________ Electrical transformer The EMF predicted by Faraday's law is also responsible for electrical transformers. When the electric current in a loop of wire changes, the changing current creates a changing magnetic field. A second wire in reach of this magnetic field will experience this change in magnetic field as a change in its coupled magnetic flux, a d Φ B / d t . Therefore, an electromotive force is set up in the second loop called the induced EMF or transformer EMF . If the two ends of this loop are connected through an electrical load, current will flow. Magnetic flow meter Faraday's law is used for measuring the flow of electrically conductive liquids and slurries. Such instruments are called magnetic flow meters. The induced voltage ℇ gen-erated in the magnetic field B due to a conductive liquid moving at velocity v is thus given by: , where ℓ is the distance between electrodes in the magnetic flow meter.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
This case is called an induced EMF. On the other hand, when the magnet is stationary and the conductor is rotated, the moving charges experience a magnetic force (as described by the Lorentz force law), and this magnetic force pushes the charges through the wire. This case is called motional EMF. Electrical motor An electrical generator can be run backwards to become a motor. For example, with the Faraday disc, suppose a DC current is driven through the conducting radial arm by a voltage. Then by the Lorentz force law, this traveling charge experiences a force in the magnetic field B that will turn the disc in a direction given by Fleming's left hand rule. In the absence of irreversible effects, like friction or Joule heating, the disc turns at the rate necessary to make d Φ B / dt equal to the voltage driving the current. Electrical transformer The EMF predicted by Faraday's law is also responsible for electrical transformers. When the electric current in a loop of wire changes, the changing current creates a changing ________________________ WORLD TECHNOLOGIES ________________________ magnetic field. A second wire in reach of this magnetic field will experience this change in magnetic field as a change in its coupled magnetic flux, a d Φ B / d t . Therefore, an electromotive force is set up in the second loop called the induced EMF or transformer EMF . If the two ends of this loop are connected through an electrical load, current will flow. Magnetic flow meter Faraday's law is used for measuring the flow of electrically conductive liquids and slurries. Such instruments are called magnetic flow meters. The induced voltage ℇ generated in the magnetic field B due to a conductive liquid moving at velocity v is thus given by: , where ℓ is the distance between electrodes in the magnetic flow meter.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
Since we already know the induced emf, we can connect these two expressions by Faraday’s law to solve for the induced electric field. Solution The induced electric field in the coil is constant in magnitude over the cylindrical surface, similar to how Ampere’s law problems with cylinders are solved. Since E → is tangent to the coil, ∮ E → · d l → = ∮ Edl = 2πrE. When combined with Equation 13.12, this gives E = ε 2πr . The direction of ε is counterclockwise, and E → circulates in the same direction around the coil. The values of E are E(t 1 ) = 6.0 V 2π (0.50 m) = 1.9 V/m; E(t 2 ) = 4.7 V 2π (0.50 m) = 1.5 V/m; E(t 3 ) = 0.040 V 2π (0.50 m) = 0.013 V/m. Significance When the magnetic flux through a circuit changes, a nonconservative electric field is induced, which drives current through the circuit. But what happens if dB/dt ≠ 0 in free space where there isn’t a conducting path? The answer is that this case can be treated as if a conducting path were present; that is, nonconservative electric fields are induced wherever dB/dt ≠ 0, whether or not there is a conducting path present. These nonconservative electric fields always satisfy Equation 13.12. For example, if the circular coil of Figure 13.9 were removed, an electric field in free space at r = 0.50 m would still be directed counterclockwise, and its Chapter 13 | Electromagnetic Induction 599 magnitude would still be 1.9 V/m at t = 0 , 1.5 V/m at t = 5.0 × 10 −2 s, etc. The existence of induced electric fields is certainly not restricted to wires in circuits. Example 13.8 Electric Field Induced by the Changing Magnetic Field of a Solenoid Part (a) of Figure 13.18 shows a long solenoid with radius R and n turns per unit length; its current decreases with time according to I = I 0 e −αt . What is the magnitude of the induced electric field at a point a distance r from the central axis of the solenoid (a) when r > R and (b) when r < R [see part (b) of Figure 13.18].
- J. A. Brandão Faria(Author)
- 2008(Publication Date)
- Wiley(Publisher)
Since the charged particle is at rest v = 0 with respect to O, this observer interprets its trajectory change as the result of a purely electric force F = Q E . In order to make both observations agree we have to conclude that E = E + v × B E v (5.17) where E v is the so-called dynamic electric field. Although we are not going to prove it, B could be obtained as B = B − v × E /c 2 . Therefore, for typical applications c B = B . The induction law, for bodies at rest subjected to time-varying magnetic fields, has been formulated in (5.9) as S E g + E i · d s = − d dt S s B t · n S dS For moving bodies subjected to stationary magnetic fields, the above equation should be modified to S t E g + E v · d s = − d dt S s t B · n S t dS (5.18) where the circulation path moves with the moving parts of the circuit, s = s t . In the most general case of moving bodies subjected to time-varying magnetic fields, the generalization of the Maxwell–Faraday induction law takes the form S t E g + E i + E v E · d s = − d dt S s t B t · n S t dS s t (5.19) where, it should be stressed, the E field on the left-hand side refers to the electric field as observed in the moving reference frame. Magnetic Induction Phenomena 221 5.10 Application Example (Electromechanical Energy Conversion) We now present an example that illustrates the principle of conversion of mechanical energy into electric energy and the conversion of electric energy into mechanical energy. Take the situation depicted in Figure 5.13 where a moving bar, of mass M and internal resistance R , slides (frictionless) with velocity v = v t e x over two conducting rails. Perpendicular to the plane defined by the two rails, a uniform time-invariant B field is enforced. Neglect the internal resistances of the rails as well as the magnetic field produced by the circulating current.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
550 Chapter 22 | Electromagnetic Induction 22.1 | Induced Emf and Induced Current There are a number of ways a magnetic field can be used to generate an electric current, and Figure 22.1 illustrates one of them. This drawing shows a bar magnet and a helical coil of wire to which an ammeter is connected. When there is no relative motion between the magnet and the coil, as in part a of the drawing, the ammeter reads zero, indicating that no current exists. However, when the magnet moves toward the coil, as in part b, a current I appears. As the magnet approaches, the magnetic field B B that it creates at the location of the coil becomes stronger and stronger, and it is this changing field that produces the current. When the magnet moves away from the coil, as in part c, a current is also pro- duced, but with a reversed direction. Now the magnetic field at the coil becomes weaker as the magnet moves away. Once again it is the changing field that generates the current. A current would also be created in Figure 22.1 if the magnet were held stationary and the coil were moved, because the magnetic field at the coil would be changing as the coil approached or receded from the magnet. Only relative motion between the magnet and the coil is needed to generate a current; it does not matter which one moves. The current in the coil is called an induced current because it is brought about (or “in- duced”) by a changing magnetic field. Since a source of emf (electromotive force) is always needed to produce a current, the coil itself behaves as if it were a source of emf. This emf is known as an induced emf. Thus, a changing magnetic field induces an emf in the coil, and the emf leads to an induced current. 22 | Electromagnetic Induction © Jeff Greenberg/Age Fotostock Electric guitars are famous for their amplified and manipulatable sound. To produce this sound, virtually all of them have one or more electro- magnetic pickups located beneath the strings (see Section 22.6).
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Since Faraday investigated electro- magnetic induction in more detail and published his findings first, the law that describes the phenomenon bears his name. Faraday discovered that whenever there is a change in flux through a loop of wire, an emf is induced in the loop. In this context, the word ‘change’ refers to a change as time passes. A flux that is constant in time creates no emf. Faraday’s law of electromagnetic induction is expressed by bringing together the idea of magnetic flux and the time interval during which it changes. In fact, Faraday found that the magnitude of the induced emf is equal to the time rate of change of the magnetic flux. This is consistent with the relation we obtained in section 22.3 for the specific case of motional emf: ℰ = −ΔΦ/Δt. CHAPTER 22 Electromagnetic induction 621 Often the magnetic flux passes through a coil of wire containing more than one loop (or turn). If the coil consists of N loops, and if the same flux passes through each loop, it is found experimentally that the total induced emf is N times that induced in a single loop. An analogous situation occurs in a flashlight when two 1.5‐V batteries are stacked in series on top of one another to give a total emf of 3.0 volts. For the general case of N loops, the total induced emf is described by Faraday’s law of electromagnetic induction in the following manner. Faraday’s law of electromagnetic induction The average emf ℰ induced in a coil of N loops is ℰ = -N ( Φ - Φ 0 t - t 0 ) = -N ΔΦ Δt (22.3) where ΔΦ is the change in magnetic flux through one loop and Δt is the time interval during which the change occurs. The term ΔΦ/Δt is the average time rate of change of the flux that passes through one loop. SI unit of induced emf: volt (V) Faraday’s law states that an emf is generated if the magnetic flux changes for any reason. Since the flux is given by equation 22.2 as Φ = BA cos , it depends on the three factors, B, A, and , any of which may change.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
LEARNING OBJECTIVES After reading this module, you should be able to... 22.1 Predict when an induced current will flow. 22.2 Solve motional emf problems. 22.3 Calculate magnetic flux. 22.4 Solve problems using Faraday’s law of induction. 22.5 Predict the direction of an induced current using Lenz’s law. 22.6 Describe how sound is reproduced via induction. 22.7 Solve problems involving generators. 22.8 Define mutual induction and self- inductance. 22.9 Solve problems involving transformers. © Jeff Greenberg/Age Fotostock CHAPTER 22 Electromagnetic Induction Electric guitars are famous for their amplified and manipulatable sound. To produce this sound, virtually all of them have one or more electromagnetic pickups located beneath the strings (see Section 22.6). These pickups work because of electromagnetic induction, which is the process by which a magnet is used to create or induce an emf in a coil of wire. In this photograph the pickup is indicated by the shiny rectangle in the white area. 22.1 Induced Emf and Induced Current There are a number of ways a magnetic field can be used to generate an electric cur- rent, and Interactive Figure 22.1 illustrates one of them. This drawing shows a bar magnet and a helical coil of wire to which an ammeter is connected. When there is no relative motion between the magnet and the coil, as in part a of the drawing, the ammeter reads zero, indicating that no current exists. However, when the magnet moves toward the coil, as in part b, a current I appears. As the magnet approaches, the magnetic field B → that it creates at the location of the coil becomes stronger and stron- ger, and it is this changing field that produces the current. When the magnet moves away from the coil, as in part c, a current is also produced, but with a reversed direc- tion. Now the magnetic field at the coil becomes weaker as the magnet moves away. Once again it is the changing field that generates the current.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
As each tape magnet goes by the gap, some magnetic field lines pass through the core and coil. The changing flux in the coil creates an induced emf. The gap width has been exaggerated. Figure 22.20 Electric generators such as these supply electric power by producing an induced emf according to Faraday’s law of electromagnetic induction. David Weintraub/Science Source 632 Chapter 22 | Electromagnetic Induction of RHR-1 (fingers of extended right hand point along B B , thumb along the velocity v B , palm pushes in the direction of the force on a positive charge), it can be seen that the direction of the current is from bottom to top in the left side and from top to bottom in the right side. Thus, charge flows around the loop. The upper and lower segments of the loop are also moving. However, these segments can be ignored because the magnetic force on the charges within them points toward the sides of the wire and not along the length. The magnitude of the motional emf developed in a conductor moving through a mag- netic field is given by Equation 22.1. To apply this expression to the left side of the coil, whose length is L (see Figure 22.21c), we need to use the velocity component v ' that is perpendicular to B B . Letting u be the angle between v B and B B (see Figure 22.21b), it follows that v ' 5 v sin u, and, with the aid of Equation 22.1, the emf can be written as % 5 BLv ' 5 BLv sin u The emf induced in the right side has the same magnitude as that in the left side. Since the emfs from both sides drive current in the same direction around the loop, the emf for the complete loop is % 5 2BLv sin u. If the coil consists of N loops, the net emf is N times as great as that of one loop, so % 5 N(2BLv sin u) It is convenient to express the variables v and u in terms of the angular speed v at which the coil rotates. Equation 8.2 shows that the angle u is the product of the angular speed and the time, u 5 vt, if it is assumed that u 5 0 rad when t 5 0 s.
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