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Ampere’s Law

Ampere’s Law. General Statement. Magnetic fields add as vectors, currents – as scalars. Just as with the integral form of Gauss’s law, the integral form of Ampere’s law is powerful to use in symmetric situations. Magnetic Field of a Solenoid. Field of a toroidal solenoid.

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Ampere’s Law

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  1. Ampere’s Law

  2. General Statement Magnetic fields add as vectors, currents – as scalars

  3. Just as with the integral form of Gauss’s law, the integral form of Ampere’s law is powerful to use in symmetric situations

  4. Magnetic Field of a Solenoid

  5. Field of a toroidal solenoid

  6. Magnetic Field of a Sheet of Current The field of a magnetic “capacitor”

  7. Magnetic materials When materials are placed in a magnetic field, they get magnetized. In majority of materials, the magnetic effects are small. Some however show strong responses. • The small magnetism is of two kinds: • Diamagnetics are repelled from magnetic fields • Paramagnetics are attracted towards magnetic fields • This is unlike the electric effect in matter, which always causes dielectrics to be attracted.

  8. The Bohr Magnetron

  9. Magnetization

  10. Diamagnetism occurs in substances where magnetic moments inside atoms all cancel out, the net magnetic moment of the atom is zero. The induced magnetic moment is directed opposite to the applied field. Diamagnetism is weakly dependent on T. • Diamagnetic (induced atomic moment) effect is overcome in paramagnetic materials, whose atoms have uncompensated magnetic moments. These moments align with the applied field to enhance the latter. Temperature T wants to destroy alignment, hence a strong (1/T) dependence. Magnetic effects are a completely quantum-mechanical phenomenon, although some classical physics arguments can be made.

  11. Example: Magnetic dipoles in a paramagnetic material Nitric oxide (NO) is a paramagnetic compound. Its molecules have maximum magnetic moment of ~ mB . In a magnetic field B=1.5 Tesla, compare the interaction energy of the magnetic moments with the field to the average translational kinetic energy of the molecules at T=300 K.

  12. Alignment of magnetic domains in applied field Ferromagnetism • In ferromagnetic materials, in addition to atoms having uncompensated magnetic moments, these moments strongly interact between themselves. • Strongly nonlinear behavior with remnant magnetization left when the applied field is lifted. Permeability Km is much larger, ~1,000 to 100,000

  13. Magnetization curve for soft iron showing hysteresis Hysteresis and Permanent Magnets Magnetization value depends on the “history” of applied magnetic field Example: A ferromagnetic material A permanent magnet is made of a ferromagnetic material with a M~106 A/m The magnet is in the shape of a cube of side 2 cm. Find magnetic dipole moment of a magnet. Estimate the magnetic field at a point 10 cm away on the axis

  14. Experiments leading to Faraday’s Law Electromagnetic Induction – Time-varying magnetic field creates electric field

  15. Changing Magnetic Flux No current in the electromagnet – B=0 - galvanometer shows no current. When magnet is turned on – momentarily current appears as B increases. When B reaches steady value – current disappears no matter how strong B field is. If we squeeze the coil as to change its area – current appears but only while we are deforming the coil. If we rotate the coil, current appears but only while we are rotating it. If we start displacing the coil out of the magnetic field – current appears while the coil is in motion. If we decrease/increase the number of loops in the coil – current appears during winding/unwinding of the turns. If we turn off the magnet – current appears while the magnetic field is being disappearing The faster we carry out all those changes - the greater the current is.

  16. Faraday’s Law quantified

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