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

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

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 point P

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

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

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

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:

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:

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