Chapter 19

1. Chapter 19 Magnetism

2. Magnetic Fields II Sections 6–10

3. Motion of a Charged Particle in a Uniform Magnetic Field • Consider a particle moving in an external magnetic field so that its velocity is perpendicular to the field • The force is always directed toward the center of the circular path • The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle

4. Motion of a Charged Particle in a Uniform Magnetic Field, cont • Equating the magnetic and centripetal forces: • Solving for the radius r: • r is proportional to the momentum mv of the particle and inversely proportional to the magnetic field • Sometimes called the cyclotron equation Active Figure: Motion of a Charged Particle in a Uniform Magnetic Field

5. The Mass Spectrometer: Separating Isotopes • The cyclotron equation can be applied to the process of separating isotopes • Singly ionized isotopes are injected into a velocity selector • Only those isotopes with velocity v = E/B pass into the deflection chamber—Why? • Isotopes travel in different circular paths governed by the cyclotron equation—therefore different mass isotopes separate Active Figure: The Mass Spectrometer

6. Particle Moving in an External Magnetic Field • If the particle’s velocity is not perpendicular to the magnetic field, the path followed by the particle is a spiral • The spiral path is called a helix Active Figure: A Charged Particle with a Helical Path

7. Charged Particles Trapped in the Earth’s Magnetic Field—Auroras • Charged particles from the Sun enter the Earth’s magnetic field • These particles move in spirals around the lines of magnetic field • This causes them to become trapped in the Earth’s magnetic field • An aurora is caused by these trapped charged particles colliding with atoms in the upper atmosphere—producing beautiful displays of light

8. Hans Christian Oersted • 1777 – 1851 • Best known for observing that a compass needle deflects when placed near a wire carrying a current • First evidence of a connection between electric and magnetic phenomena

9. Magnetic Fields – Long Straight Wire • A current-carrying wire produces a magnetic field • The compass needle deflects in directions tangent to the circle • The compass needle points in the direction of the magnetic field produced by the current Active Figure: Magnetic Field Due to a Long Straight Wire

10. Direction of the Field of a Long Straight Wire • Right Hand Rule #2 • Grasp the wire in your right hand • Point your thumb in the direction of the current • Your fingers will curl in the direction of the field

11. Magnitude of the Field of a Long Straight Wire • The magnitude of the field at a distance r from a wire carrying a current of I is • µo = 4  x 10-7 T.m / A • µo is called the permeability of free space

12. André-Marie Ampère • 1775 – 1836 • Credited with the discovery of electromagnetism • Relationship between electric currents and magnetic fields • Mathematical genius evident by age 12

13. Ampère’s Law • André-Marie Ampère found a procedure for deriving the relationship between the current in a wire and the magnetic field produced by the wire • Ampère’s Circuital Law • B|| Δℓ = µo I • Sum over the closed path around the current I • Choose an arbitrary closed path around the current • Sum all the products of B|| Δℓ around the closed path

14. Ampère’s Law to Find B for a Long Straight Wire • Sum over a closed circular path around current I B|| Δℓ = µo I • Sum all products B|| Δℓ around the closed path B·2r = µo I • The magnitude of the magnetic field a distance r from the wire

15. Magnetic Field of a Current Loop • The strength of a magnetic field produced by a wire can be enhanced by forming the wire into a loop • All the segments, Δx, contribute to the field, increasing its strength • The magnitude of the magnetic field at the center of a circular loop with a radius R

16. Magnetic Field of a Current Loop – Total Field

17. Magnetic Field of a Solenoid • If a long straight wire is bent into a coil of several closely spaced loops, the resulting device is called a solenoid • It is also known as an electromagnet since it acts like a magnet only when it carries a current

18. Magnetic Field of a Solenoid, 2 • The field lines inside the solenoid are nearly parallel, uniformly spaced, and close together • This indicates that the field inside the solenoid is nearly uniform and strong • The exterior field is nonuniform, much weaker, and in the opposite direction to the field inside the solenoid

19. Magnetic Field in a Solenoid, 3 • The field lines of the solenoid resemble those of a bar magnet – dipole magnetic field

20. Magnetic Field in a Solenoid from Ampère’s Law • A cross-sectional view of a tightly wound solenoid • If the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero • Apply Ampère’s Law to the blue dashed rectangle • The magnitude of the field inside a solenoid is constant at all points far from its ends • n is the number of turns per unit length • n = N / ℓ

21. Magnetic Force Between Two Parallel Conductors • The force on wire 1 is due to the current in wire 1 and the magnetic field produced by wire 2 • The force per unit length is:

22. Force Between Two Conductors, cont • Parallel conductors carrying currents in the same direction attract each other • Parallel conductors carrying currents in the opposite directions repel each other Active Figure: Force Between Long Parallel Wires

23. Defining Ampere and Coulomb • The force between parallel conductors can be used to define the Ampere (A) • If two long, parallel wires 1 m apart carry the same current, and the magnitude of the magnetic force per unit length is 2 x 10-7 N/m, then the current is defined to be 1 A • The SI unit of charge, the Coulomb (C), can be defined in terms of the Ampere • If a conductor carries a steady current of 1 A, then the quantity of charge that flows through any cross section in 1 second is 1 C

24. Magnetic Effects of Electrons – Orbits • An individual atom should act like a magnet because of the motion of the electrons about the nucleus • Each electron circles the atom once in about every 10-16 seconds • This would produce a current of 1.6 mA and a magnetic field of about 20 T at the center of the circular path • However, the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving electron in the same atom • The net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials

25. Magnetic Effects of Electrons – Spins • Electrons also have spin • The classical model is to consider the electrons to spin like tops • It is actually a quantum effect

26. Magnetic Effects of Electrons – Spins, cont • The field due to the spinning is generally stronger than the field due to the orbital motion • Electrons usually pair up with their spins opposite each other, so their fields cancel each other • That is why most materials are not naturally magnetic

27. Magnetic Effects of Electrons – Domains • In some materials, the spins do not naturally cancel • Such materials are called ferromagnetic • Large groups of atoms in which the spins are aligned are called domains • When an external field is applied, the domains that are aligned with the field tend to grow at the expense of the others • This causes the material to become magnetized

28. Domains, cont • Random alignment (left) shows an unmagnetized material • When an external field is applied, the domains aligned with B grow (right)

29. Domains and Permanent Magnets • In hard magnetic materials, the domains remain aligned after the external field is removed • The result is a permanent magnet • In soft magnetic materials, once the external field is removed, thermal agitation causes the materials to quickly return to an unmagnetized state • When a ferromagnetic core is placed inside a current-carrying loop, the magnetic field is enhanced since the domains in the core material align, increasing the magnetic field