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University Physics Volume 2

12.5 Ampère’s Law

University Physics Volume 212.5 Ampère’s Law
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  1. Preface
  2. Unit 1. Thermodynamics
    1. 1 Temperature and Heat
      1. Introduction
      2. 1.1 Temperature and Thermal Equilibrium
      3. 1.2 Thermometers and Temperature Scales
      4. 1.3 Thermal Expansion
      5. 1.4 Heat Transfer, Specific Heat, and Calorimetry
      6. 1.5 Phase Changes
      7. 1.6 Mechanisms of Heat Transfer
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 The Kinetic Theory of Gases
      1. Introduction
      2. 2.1 Molecular Model of an Ideal Gas
      3. 2.2 Pressure, Temperature, and RMS Speed
      4. 2.3 Heat Capacity and Equipartition of Energy
      5. 2.4 Distribution of Molecular Speeds
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 The First Law of Thermodynamics
      1. Introduction
      2. 3.1 Thermodynamic Systems
      3. 3.2 Work, Heat, and Internal Energy
      4. 3.3 First Law of Thermodynamics
      5. 3.4 Thermodynamic Processes
      6. 3.5 Heat Capacities of an Ideal Gas
      7. 3.6 Adiabatic Processes for an Ideal Gas
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 The Second Law of Thermodynamics
      1. Introduction
      2. 4.1 Reversible and Irreversible Processes
      3. 4.2 Heat Engines
      4. 4.3 Refrigerators and Heat Pumps
      5. 4.4 Statements of the Second Law of Thermodynamics
      6. 4.5 The Carnot Cycle
      7. 4.6 Entropy
      8. 4.7 Entropy on a Microscopic Scale
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Electricity and Magnetism
    1. 5 Electric Charges and Fields
      1. Introduction
      2. 5.1 Electric Charge
      3. 5.2 Conductors, Insulators, and Charging by Induction
      4. 5.3 Coulomb's Law
      5. 5.4 Electric Field
      6. 5.5 Calculating Electric Fields of Charge Distributions
      7. 5.6 Electric Field Lines
      8. 5.7 Electric Dipoles
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Gauss's Law
      1. Introduction
      2. 6.1 Electric Flux
      3. 6.2 Explaining Gauss’s Law
      4. 6.3 Applying Gauss’s Law
      5. 6.4 Conductors in Electrostatic Equilibrium
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 7 Electric Potential
      1. Introduction
      2. 7.1 Electric Potential Energy
      3. 7.2 Electric Potential and Potential Difference
      4. 7.3 Calculations of Electric Potential
      5. 7.4 Determining Field from Potential
      6. 7.5 Equipotential Surfaces and Conductors
      7. 7.6 Applications of Electrostatics
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Capacitance
      1. Introduction
      2. 8.1 Capacitors and Capacitance
      3. 8.2 Capacitors in Series and in Parallel
      4. 8.3 Energy Stored in a Capacitor
      5. 8.4 Capacitor with a Dielectric
      6. 8.5 Molecular Model of a Dielectric
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 9 Current and Resistance
      1. Introduction
      2. 9.1 Electrical Current
      3. 9.2 Model of Conduction in Metals
      4. 9.3 Resistivity and Resistance
      5. 9.4 Ohm's Law
      6. 9.5 Electrical Energy and Power
      7. 9.6 Superconductors
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Direct-Current Circuits
      1. Introduction
      2. 10.1 Electromotive Force
      3. 10.2 Resistors in Series and Parallel
      4. 10.3 Kirchhoff's Rules
      5. 10.4 Electrical Measuring Instruments
      6. 10.5 RC Circuits
      7. 10.6 Household Wiring and Electrical Safety
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Magnetic Forces and Fields
      1. Introduction
      2. 11.1 Magnetism and Its Historical Discoveries
      3. 11.2 Magnetic Fields and Lines
      4. 11.3 Motion of a Charged Particle in a Magnetic Field
      5. 11.4 Magnetic Force on a Current-Carrying Conductor
      6. 11.5 Force and Torque on a Current Loop
      7. 11.6 The Hall Effect
      8. 11.7 Applications of Magnetic Forces and Fields
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 12 Sources of Magnetic Fields
      1. Introduction
      2. 12.1 The Biot-Savart Law
      3. 12.2 Magnetic Field Due to a Thin Straight Wire
      4. 12.3 Magnetic Force between Two Parallel Currents
      5. 12.4 Magnetic Field of a Current Loop
      6. 12.5 Ampère’s Law
      7. 12.6 Solenoids and Toroids
      8. 12.7 Magnetism in Matter
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    9. 13 Electromagnetic Induction
      1. Introduction
      2. 13.1 Faraday’s Law
      3. 13.2 Lenz's Law
      4. 13.3 Motional Emf
      5. 13.4 Induced Electric Fields
      6. 13.5 Eddy Currents
      7. 13.6 Electric Generators and Back Emf
      8. 13.7 Applications of Electromagnetic Induction
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 14 Inductance
      1. Introduction
      2. 14.1 Mutual Inductance
      3. 14.2 Self-Inductance and Inductors
      4. 14.3 Energy in a Magnetic Field
      5. 14.4 RL Circuits
      6. 14.5 Oscillations in an LC Circuit
      7. 14.6 RLC Series Circuits
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 15 Alternating-Current Circuits
      1. Introduction
      2. 15.1 AC Sources
      3. 15.2 Simple AC Circuits
      4. 15.3 RLC Series Circuits with AC
      5. 15.4 Power in an AC Circuit
      6. 15.5 Resonance in an AC Circuit
      7. 15.6 Transformers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 16 Electromagnetic Waves
      1. Introduction
      2. 16.1 Maxwell’s Equations and Electromagnetic Waves
      3. 16.2 Plane Electromagnetic Waves
      4. 16.3 Energy Carried by Electromagnetic Waves
      5. 16.4 Momentum and Radiation Pressure
      6. 16.5 The Electromagnetic Spectrum
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Explain how Ampère’s law relates the magnetic field produced by a current to the value of the current
  • Calculate the magnetic field from a long straight wire, either thin or thick, by Ampère’s law

A fundamental property of a static magnetic field is that, unlike an electrostatic field, it is not conservative. A conservative field is one that does the same amount of work on a particle moving between two different points regardless of the path chosen. Magnetic fields do not have such a property. Instead, there is a relationship between the magnetic field and its source, electric current. It is expressed in terms of the line integral of BB and is known as Ampère’s law. This law can also be derived directly from the Biot-Savart law. We now consider that derivation for the special case of an infinite, straight wire.

Figure 12.14 shows an arbitrary plane perpendicular to an infinite, straight wire whose current I is directed out of the page. The magnetic field lines are circles directed counterclockwise and centered on the wire. To begin, let’s consider B·dlB·dl over the closed paths M and N. Notice that one path (M) encloses the wire, whereas the other (N) does not. Since the field lines are circular, B·dlB·dl is the product of B and the projection of dl onto the circle passing through dl.dl. If the radius of this particular circle is r, the projection is rdθ,rdθ, and

B·dl=Brdθ.B·dl=Brdθ.
Figures A and B show an arbitrary plane perpendicular to an infinite, straight wire whose current I is directed out of the page. The magnetic field lines are circles directed counterclockwise and centered on the wire. Ampere path M demonstrated in the Figure A encloses the wire. Ampere path N demonstrated in the Figure B does not enclose the wire.
Figure 12.14 The current I of a long, straight wire is directed out of the page. The integral dθdθ equals 2π2π and 0, respectively, for paths M and N.

With BB given by Equation 12.9,

B·dl=(μ0I2πr)rdθ=μ0I2πdθ.B·dl=(μ0I2πr)rdθ=μ0I2πdθ.
(12.20)

For path M, which circulates around the wire, Mdθ=2πMdθ=2π and

MB·dl=μ0I.MB·dl=μ0I.
(12.21)

Path N, on the other hand, circulates through both positive (counterclockwise) and negative (clockwise) dθdθ (see Figure 12.14), and since it is closed, Ndθ=0.Ndθ=0. Thus for path N,

NB·dl=0.NB·dl=0.
(12.22)

The extension of this result to the general case is Ampère’s law.

Ampère’s law

Over an arbitrary closed path,

B·dl=μ0IB·dl=μ0I
(12.23)

where I is the total current passing through any open surface S whose perimeter is the path of integration. Only currents inside the path of integration need be considered.

To determine whether a specific current I is positive or negative, curl the fingers of your right hand in the direction of the path of integration, as shown in Figure 12.14. If I passes through S in the same direction as your extended thumb, I is positive; if I passes through S in the direction opposite to your extended thumb, it is negative.

Problem-Solving Strategy: Ampère’s Law

To calculate the magnetic field created from current in wire(s), use the following steps:

  1. Identify the symmetry of the current in the wire(s). If there is no symmetry, use the Biot-Savart law to determine the magnetic field.
  2. Determine the direction of the magnetic field created by the wire(s) by right-hand rule 2.
  3. Chose a path loop where the magnetic field is either constant or zero.
  4. Calculate the current inside the loop.
  5. Calculate the line integral B·dlB·dl around the closed loop.
  6. Equate B·dlB·dl with μ0Iencμ0Ienc and solve for B.B.

Example 12.6

Using Ampère’s Law to Calculate the Magnetic Field Due to a Wire Use Ampère’s law to calculate the magnetic field due to a steady current I in an infinitely long, thin, straight wire as shown in Figure 12.15.

Figures shows an infinitely long, thin, straight wire with the current directed out of the page. The possible magnetic field components in this plane, BR and BTheta, are shown at arbitrary points on a circle of radius r centered on the wire.
Figure 12.15 The possible components of the magnetic field B due to a current I, which is directed out of the page. The radial component is zero because the angle between the magnetic field and the path is at a right angle.

Strategy Consider an arbitrary plane perpendicular to the wire, with the current directed out of the page. The possible magnetic field components in this plane, BrBr and Bθ,Bθ, are shown at arbitrary points on a circle of radius r centered on the wire. Since the field is cylindrically symmetric, neither BrBr nor BθBθ varies with the position on this circle. Also from symmetry, the radial lines, if they exist, must be directed either all inward or all outward from the wire. This means, however, that there must be a net magnetic flux across an arbitrary cylinder concentric with the wire. The radial component of the magnetic field must be zero because Brdl=0.Brdl=0. Therefore, we can apply Ampère’s law to the circular path as shown.

Solution Over this path BB is constant and parallel to dl,dl, so

B·dl=Bθdl=Bθ(2πr).B·dl=Bθdl=Bθ(2πr).

Thus Ampère’s law reduces to

Bθ(2πr)=μ0I.Bθ(2πr)=μ0I.

Finally, since BθBθ is the only component of B,B, we can drop the subscript and write

B=μ0I2πr.B=μ0I2πr.

This agrees with the Biot-Savart calculation above.

Significance Ampère’s law works well if you have a path to integrate over which B·dlB·dl has results that are easy to simplify. For the infinite wire, this works easily with a path that is circular around the wire so that the magnetic field factors out of the integration. If the path dependence looks complicated, you can always go back to the Biot-Savart law and use that to find the magnetic field.

Example 12.7

Calculating the Magnetic Field of a Thick Wire with Ampère’s Law The radius of the long, straight wire of Figure 12.16 is a, and the wire carries a current I0I0 that is distributed uniformly over its cross-section. Find the magnetic field both inside and outside the wire.

Figure A shows a long, straight wire of radius a that carries current I. Figure B shows a cross-section of the same wire with the Ampère’s loop of radius r.
Figure 12.16 (a) A model of a current-carrying wire of radius a and current I0.I0. (b) A cross-section of the same wire showing the radius a and the Ampère’s loop of radius r.

Strategy This problem has the same geometry as Example 12.6, but the enclosed current changes as we move the integration path from outside the wire to inside the wire, where it doesn’t capture the entire current enclosed (see Figure 12.16).

Solution For any circular path of radius r that is centered on the wire,

B·dl=Bdl=Bdl=B(2πr).B·dl=Bdl=Bdl=B(2πr).

From Ampère’s law, this equals the total current passing through any surface bounded by the path of integration.

Consider first a circular path that is inside the wire (ra)(ra) such as that shown in part (a) of Figure 12.16. We need the current I passing through the area enclosed by the path. It’s equal to the current density J times the area enclosed. Since the current is uniform, the current density inside the path equals the current density in the whole wire, which is I0/πa2.I0/πa2. Therefore the current I passing through the area enclosed by the path is

I=πr2πa2I0=r2a2I0.I=πr2πa2I0=r2a2I0.

We can consider this ratio because the current density J is constant over the area of the wire. Therefore, the current density of a part of the wire is equal to the current density in the whole area. Using Ampère’s law, we obtain

B(2πr)=μ0(r2a2)I0,B(2πr)=μ0(r2a2)I0,

and the magnetic field inside the wire is

B=μ0I02πra2(ra).B=μ0I02πra2(ra).

Outside the wire, the situation is identical to that of the infinite thin wire of the previous example; that is,

B=μ0I02πr(ra).B=μ0I02πr(ra).

The variation of B with r is shown in Figure 12.17.

Graph shows the variation of B with r. It linearly increases with the r until the point a. Then it starts to decrease proportionally to the inverse of r.
Figure 12.17 Variation of the magnetic field produced by a current I0I0 in a long, straight wire of radius a.

Significance The results show that as the radial distance increases inside the thick wire, the magnetic field increases from zero to a familiar value of the magnetic field of a thin wire. Outside the wire, the field drops off regardless of whether it was a thick or thin wire.

This result is similar to how Gauss’s law for electrical charges behaves inside a uniform charge distribution, except that Gauss’s law for electrical charges has a uniform volume distribution of charge, whereas Ampère’s law here has a uniform area of current distribution. Also, the drop-off outside the thick wire is similar to how an electric field drops off outside of a linear charge distribution, since the two cases have the same geometry and neither case depends on the configuration of charges or currents once the loop is outside the distribution.

Example 12.8

Using Ampère’s Law with Arbitrary Paths Use Ampère’s law to evaluate B·dlB·dl for the current configurations and paths in Figure 12.18.

Figure A shows four wires carrying currents of two Amperes, five Amperes, three Amperes, and four Amperes. All four wires are inside the loop. First and second wires carry current downward through the loop. Third and fourth wires carry current upward through the loop. Figure B shows three wires carrying currents of five Amperes, two Amperes, and three Amperes. First and third wires are outside the loop, second wire is inside the loop. First wire carries current upward through the loop. Second and third wires carry current downward through the loop. Figure C shows three wires carrying currents of seven Amperes, five Amperes, and three Amperes. All three wires are inside the loop. First and second wires carry current downward through the loop. Third wire carries current upward through the loop.
Figure 12.18 Current configurations and paths for Example 12.8.

Strategy Ampère’s law states that B·dl=μ0IB·dl=μ0I where I is the total current passing through the enclosed loop. The quickest way to evaluate the integral is to calculate μ0Iμ0I by finding the net current through the loop. Positive currents flow with your right-hand thumb if your fingers wrap around in the direction of the loop. This will tell us the sign of the answer.

Solution (a) The current going downward through the loop equals the current going out of the loop, so the net current is zero. Thus, B·dl=0.B·dl=0.

(b) The only current to consider in this problem is 2A because it is the only current inside the loop. The right-hand rule shows us the current going downward through the loop is in the positive direction. Therefore, the answer is B·dl=μ0(2A)=2.51×10−6Tm.B·dl=μ0(2A)=2.51×10−6Tm.

(c) The right-hand rule shows us the current going downward through the loop is in the positive direction. There are 7A+5A=12A7A+5A=12A of current going downward and –3 A going upward. Therefore, the total current is 9 A and B·dl=μ0(9A)=5.65×10−6Tm.B·dl=μ0(9A)=5.65×10−6Tm.

Significance If the currents all wrapped around so that the same current went into the loop and out of the loop, the net current would be zero and no magnetic field would be present. This is why wires are very close to each other in an electrical cord. The currents flowing toward a device and away from a device in a wire equal zero total current flow through an Ampère loop around these wires. Therefore, no stray magnetic fields can be present from cords carrying current.

Check Your Understanding 12.6

Consider using Ampère’s law to calculate the magnetic fields of a finite straight wire and of a circular loop of wire. Why is it not useful for these calculations?

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