1. Jean-Baptiste Biot (1774-1862)
Physics 2113
Jonathan Dowling
Lecture 26: MON 27 OCT Magnetic Fields Due to Currents: Biot-Savart Law
Felix Savart
(1791–1841)
Jean-Baptiste Biot ( )

2. What Are We Going to Learn? A Road Map
Electric charge  Electric force on other electric charges  Electric field, and electric potential
Moving electric charges : current
Electronic circuit components: batteries, resistors, capacitors
Electric currents  Magnetic field  Magnetic force on moving charges
Time-varying magnetic field  Electric Field
More circuit components: inductors.
Electromagnetic waves  light waves
Geometrical Optics (light rays).
Physical optics (light waves)

3. Electric Current: A Source of Magnetic Field
Orested’s Observation : an electric current creates a magnetic field
Simple experiment: hold a current-carrying wire near a compass needle! I B
Wire with
current
INTO page B B

4. Hans Christian Oersted was a professor of science at Copenhagen University. In 1820 he arranged in his home a science demonstration to friends and students. He planned to demonstrate the heating of a wire by an electric current, and also to carry out demonstrations of magnetism, for which he provided a compass needle mounted on a wooden stand.
While performing his electric demonstration, Oersted noted to his surprise that every time the electric current was switched on, the compass needle moved. He kept quiet and finished the demonstrations, but in the months that followed worked hard trying to make sense out of the new phenomenon.

5. New Right Hand Rule! i B i Direction of B!
Point your thumb along the direction of the current in a straight wire
The magnetic field created by the current consists of circular loops directed along your curled fingers.
The magnetic field gets weaker with distance: For long wire it’s a 1/R Law!
You can apply this to ANY straight wire (even a small differential element!)
What if you have a curved wire? Break into small elements. i B
Direction of B! i

6. 29.2: Magnetic Field due to a Long Straight Wire:
Fig A right-hand rule gives the direction of the magnetic field due to a current in a wire. (a) The magnetic field B at any point to the left of the wire is perpendicular to the dashed radial line and directed into the page, in the direction of the fingertips, as indicated by the x. (b) If the current is reversed, at any point to the left is still perpendicular to the dashed radial line but now is directed out of the page, as indicated by the dot.

7. Superposition: ICPP
Magnetic fields (like electric fields) can be “superimposed” -- just do a vector sum of B from different sources
The figure shows four wires located at the 4 corners of a square. They carry equal currents in directions indicated
What is the direction of B at the center of the square?
I-OUT
I-OUT
I-IN
I-IN B

8. “Coulomb’s” Law For E-Fields
When we computed the electric field due to charges we used
Coulomb’s law. If one had a large irregular object, one broke it
into infinitesimal pieces and computed dE,
Which we write as,
If you wish to compute the magnetic field due to a current in a wire, you use the law of Biot and Savart.

9. The Biot-Savart Law For B-Fields
Jean-Baptiste
Biot ( )
Felix Savart ( ) i
Quantitative rule for computing the magnetic field from any electric current
Choose a differential element of wire of length dL and carrying a current i
The field dB from this element at a point located by the vector r is given by the Biot-Savart Law
μ0 =4π×10–7 T•m/A
(permeability constant of free space)

10. Biot-Savart Requires A Right-Hand Rule
Biot-Savart Law
for B-Fields
Coulomb Law
for E-Fields i
Biot-Savart Requires A Right-Hand Rule
Both Are 1/r2 Laws!
The has no units.

11. Biot-Savart Law An infinitely long straight wire carries a current i.
Determine the magnetic field generated at a point located at a perpendicular distance R from the wire.
Choose an element ds as shown
Biot-Savart Law: dB points INTO the page
Integrate over all such elements

12. Field of a Straight Wire

13. Is the B-Field From a Power Line Dangerous?
A power line carries a current of 500 A.
What is the magnetic field in a house located 100 m away from the power line?
= 1 mT
Recall that the earth’s magnetic field is ~10–4T = 100 mT
Probably not dangerous!

15. Biot-Savart Law
A circular arc of wire of radius R carries a current i.
What is the magnetic field at the center of the loop? i
Direction of B?? Not another right hand rule?!
TWO right hand rules!:
If your thumb points along the CURRENT, your fingers will point in the same direction as the FIELD.
If you curl our fingers around direction of CURRENT, your thumb points along FIELD!

16. ✔ ICPP: What is the direction of the B field at point C?
Example, Magnetic field at the center of a circular arc of a circle pg 768:
ICPP: What is the direction of the B field at point C? ✔

17. Example, Magnetic field at the center of a circular arc of a circle.:

18. Example, Magnetic field off to the side of two long straight currents: