2102
Jonathan Physics Dowling
Lecture 22: MON 09 MAR
Ch.29.12: Magnetic Fields Due to Currents
Biot-Savart Law
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Jean-Baptiste Felix Savart Biot (1774- (17911841)
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)
Electric Current: A Source of Magnetic Field
Observation: an electric current creates a magnetic field Simple experiment: hold a current-carrying wireB near a compass needle!
e ar . a re d an or ictu e ss p im pre his kT m e t uic eco se Q d to ed ed ne
I B Wire with current INTO page B
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
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QuickTime and a decompressor are needed to see this pict
Caseys Right Hand Rule!
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 its 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
Superposition
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 the of square?
I-OUT I-OUT
I-IN B
I-IN
Biot-Savart Law
When we computed the electric field due to charges we used oulombs law. If one had a large irregular object, one broke i nto infinitesimal pieces and computed, Which we write as,
r 1 dq dE = r 2 4 0 r
If you wish to compute the magnetic field due to a current in a wire, you use the law of Biot and Savart.
The Biot-Savart Law
Jean-Baptiste Biot (1774-1862)
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 0 =4x107 Tm/A
(permeability constant)
r dL
r r
dB
i
Felix Savart (1791-1841
r r r 0 idL r dB = 3 4 r
Biot-Savart Law for B-Fields
Coulomb Law for E-Fields
r ) r m0 idL r dB = 2 4p r
r dL
r dE =
1 dq ) r 2 4pe0 r
r r
dB
i
Biot-Savart Requires A Right-Hand Rule
) BothrAre 1/r Laws!
2
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
r r r 0 ids r dB = 3 4 r
0 ids (r sin ) dB = 3 4 r
m0i ds (r sin q ) B= 4p r3
Field of a Straight Wire
r r r 0 ids r dB = 3 4 r
sin = R / r
0 ids (r sin ) dB = 4 r3
r = ( s 2 + R 2 )1/ 2
m0i Rds m0i ...