Lecture 16! Fall 2010! R. F. O’Connell! 3:00-4:30 PM – TTh (or by appt) ! 109 Nicholson! Email: email@example.com! Jean-Baptiste Felix Savart ! Biot (1774-1862)! (1791–1841)! • 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) • Observation: an electric current creates a magnetic ﬁeld ! • Simple experiment: hold I! a current-carrying wire B! near a compass needle! ! Wire with! B! 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 trying to make sense out of the new phenomenon.! • Point your thumb along the direction of the current in a straight wire! • The magnetic ﬁeld created by the i! current consists of circular loops directed B! along your curled ﬁngers.! • The magnetic ﬁeld gets weaker with distance: For long wire it’s a 1/R Law!! Direction of B!! i! • You can apply this to ANY straight wire (even a small differential element!)! • What if you have a curved wire? Break into small elements.! • Magnetic ﬁelds (like electric I-OUT I-OUT 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! I-IN I-IN • What is the direction of B B at the center of the square?! 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,! Which we write as,! 1 dq dE = 2 r ˆ 4πε 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.! Jean-Baptiste ! Felix Savart ! Biot (1774-1862)! dL (1791-1841)! • Quantitative rule for computing i the magnetic ﬁeld from any electric current! r • Choose a differential element dB of wire of length dL and carrying a current i! • The ﬁeld dB from this element µ 0 idL × r at a point located by the dB = 3 vector r is given by the Biot- 4π r Savart Law! µ0 =4πx10–7 T•m/A (permeability constant)! Biot-Savart Law ! Coulomb Law ! for B-Fields! for E-Fields! µ0 idL × r ˆ 1 dq dB = dE = 2 ˆ r 2 4π r 4πε 0 r r r ˆ r= = dL r r i r dB Biot-Savart Requires A Both Are 1/r2 Laws!! Right-Hand Rule! ˆ The r "has no units.! • An inﬁnitely long straight wire carries a current i. ! • Determine the magnetic ﬁeld generated at a point located at a perpendicular distance R from µ 0 ids × r the wire.! dB = 3 4π r • Choose an element ds as shown! • Biot-Savart Law: dB points µ 0 ids (r sin θ ) dB = 3 INTO the page! 4π r • Integrate over all such ∞ µ 0i ds (r sin θ ) B= elements! 4π −∫∞ r3 µ 0 ids × r µ 0 ids (r sin θ ) dB = dB = 4π r 3 4π r3 sin θ = R / r r = ( s 2 + R 2 )1/ 2 ∞ ∞ µ 0i ds (r sin θ ) µ 0i Rds B= = 4π −∫∞ r 3 4π −∫ ( 2 + R 2 ) / 2 ∞ s 3 ∞ µ 0i Rds = ∫ (s 2 + R 2 )3 / 2 2π 0 ∞ µ 0iR s µ 0i = 2 2 = 2 1/ 2 2π R ( + R ) s 0 2πR A power line µ 0i B= carries a current 2πR of 500 A. ! −7 (4 π ×10 T ⋅ m / A)(500A) = 2π (100m) What is the = 1 µT magnetic ﬁeld in a Recall that the earth’s magnetic house located € field is ~10–4T = 100 µT 100 m away from the power line?! Probably not dangerous!! • A circular arc of wire of radius R carries a current i. ! i! • What is the magnetic ﬁeld at the center of the loop? ! µ 0 i ds × r Direction of B?? Not another dB = 3 right hand rule?!! 4π r TWO right hand rules!:! If your thumb points along the µ 0 idsR µ 0 iRdφ CURRENT, your ﬁngers will point dB = 3 = 2 in the same direction as the 4π R 4π R FIELD.! µ0 idφ µ0iΦ If you curl our ﬁngers around B= ∫ R = 4π R direction of CURRENT, your 4π thumb points along FIELD! ! Magnetic ﬁeld due to wire 1 ! where the wire 2 is,! µ0 I1 L! I1! I 2! B1 = 2π a F! Force on wire 2 due to this ﬁeld,! µ0 LI1 I 2 F21 = L I 2 B1 = 2π a a! eHarmony’s Rule for Currents: ! Same Currents – Attract!! Opposite Currents – Repel!! • Magnetic ﬁelds exert forces on moving charges:! • The force is perpendicular to the ﬁeld and the velocity.! • A current loop is a magnetic dipole moment. ! • Uniform magnetic ﬁelds exert torques on dipole moments. ! • Electric currents produce magnetic ﬁelds:! • To compute magnetic ﬁelds produced by currents, use Biot- Savart’s law for each element of current, and then integrate.! • Straight currents produce circular magnetic ﬁeld lines, with amplitude B=µ0i/2πr (use right hand rule for direction). ! • Circular currents produce a magnetic ﬁeld at the center (given by another right hand rule) equal to B=µ0iΦ/4πr • Wires currying currents produce forces on each other: eHarmony’s Rule: parallel currents attract, anti-parallel currents repel.!