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Unit 4 Day 9 – The Biot-Savant Law

Unit 4 Day 9 – The Biot-Savant Law. Restrictions of Ampere’s Law The Biot-Savant Law Differences Between Ampere’s Law & the Biot-Savant Law The Magnetic Field Due to a Current in a Straight Wire Magnetic Field On-Axis of a Current Loop

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Unit 4 Day 9 – The Biot-Savant Law

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  1. Unit 4 Day 9 – The Biot-Savant Law • Restrictions of Ampere’s Law • The Biot-Savant Law • Differences Between Ampere’s Law & the Biot-Savant Law • The Magnetic Field Due to a Current in a Straight Wire • Magnetic Field On-Axis of a Current Loop • Magnetic Field of a Wire Segment

  2. Restrictions of Ampere’s Law • Ampere’s Law is restricted to situations where the symmetry of the given currents allows us to easily evaluate the integral • Jean Baptiste Biot & Felix Savant overcame this limitation in 1820 by considering the current flowing in any path as many infinitesimal current elements

  3. Biot-Savant Law • The current flowing in any path, can be considered as many infinitesimal current elements, each of length dl, flowing in a magnetic field dB, at any pointP

  4. Biot-Savant Law • The magnitude of dB is: where θ is the angle between dl and r • The total magnetic field at pointP is: • This is equivalent to Coulomb’s Law written in differential form:

  5. Differences between Ampere’s law & the Biot-Savant Law • The difference between Ampere’s Law and the Biot-Savant Law is that in Ampere’s Law ( ), the magnetic field is not necessarily due only to the current enclosed by the path of integration, as Ampere suggests • In the Biot-Savant Law, dB is due entirely to the current element I·dl. To find the total B, it is necessary to include all currents

  6. Magnetic Field Due to Current in a Straight Wire • To find the magnetic field near an infinitely long, straight wire, carrying a current I, the Biot- Savant Law gives us: • The solution of this integral yields: • This is the same as Ampere’s Law

  7. The Magnetic Field On-Axis of a Current Loop • To find the magnetic field On-Axis, of a Current Loop, applying the Biot-Savant Law yields:

  8. Magnetic Field of a Loop at a Point On-Axis • By is = 0 from the symmetry of the problem • Bz is calculated using the Biot- Savant Law: • For z>>R then the above relationship simplifies to:

  9. Magnetic Field of a Wire Segment • Again: the Biot-Savant Law gives: • Solving the Integral yields:

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