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 point P

4. 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:

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: