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Chapter 18 Magnetic Field

5 th century BC. Greeks. Some rocks attract pieces of iron. These rocks are plentiful in the district of Magnesia AD Chinese. Use needs of magnetite to make compasses. 16th century Gilbert – earth is a giant magnet  rsted – magnetic needle responds to current in the wire.

A compass needle turns and points in a particular direction there is something which interacts with it Magnetic field (B): whatever it is that is detected by a compass Compass: similar to electric dipole Magnetic Field

Magnetic fields are produced by moving charges Current in a wire: convenient source of magnetic field Static equilibrium: net motion of electrons is zero Can make electric circuit with continuous motion of electrons The electron current (i) is the number of electrons per second that enter a section of a conductor. Counting electrons: complicated Indirect methods: measure magnetic field measure heating effect Electron Current Both are proportional to the electron current

We will use a magnetic compass as a detector of B. How can we be sure that it does not simply respond to electric fields? Interacts with iron, steel – even if they are neutral Unaffected by aluminum, plastic etc., though charged tapes interact with these materials Points toward North pole – electric dipole does not do that Detecting Magnetic Fields Compass needle:

Make electric circuit: What is the effect on the compass needle? What if we switch polarity? What if we run wire under compass? What if there is no current in the wire? Use short bulb The Magnetic Effects of Currents

Conclusions: A wire with no current produces no B B is perpendicular to the direction of current B under the wire is opposite to B over the wire The magnitude of B depends on the amount of current Oersted effect: discovered in 1820 by H. Ch. Ørsted How does the field around a wire look? The Magnetic Effects of Currents Hans Christian Ørsted ( )

Magnetic Field Due to Long Current-Carrying Wire

Principle of superposition: What can you say about the magnitudes of B Earth and B wire ? What if B Earth were much larger than B wire ? The Magnetic Effects of Currents The moving electrons in a wire create a magnetic field

A current-carrying wire is oriented N-S and laid on top of a compass. The compass needle points 27 o west. What is the magnitude and direction of the magnetic field created by the wire B wire if the magnetic field of Earth is B Earth = 2   T (tesla). Exercise

Biot-Savart Law for a Single Charge Electric field of a point charge: Moving charge makes a curly magnetic field: B units: T (tesla) = kg s -2 A -1 Jean-Baptiste Biot ( ) Felix Savart ( ) Nikola Tesla ( )

Calculate magnitude: Right-hand rule The Cross Product Calculate direction:

Clicker What is thedirection of < 0,0,3> x < 0,4,0>? A) +x B) –x C) +y D) –y E) zero magnitude

 a vector (arrow) is facing into the screen  a vector (arrow) is facing out of the screen v r B B B BB Two-dimensional Projections

What is B straight ahead? What if the charge is negative? Exercise

v r B1B1 v rB v r B1B1 Magnetic field depends on qv: Positive and negative charges produce the same B if moving in opposite directions at the same speed For the purpose of predicting B we can describe current flow in terms of ‘conventional current’ – positive moving charges. Moving Charge Sign Dependence + - -

A steady flow of charges in one direction will create a magnetic field. How can we cause charges to flow steadily? Need to find a way to produce and sustain E in a wire. Use battery Electron Current How can we know magnitude and direction of magnetic field produced by wire? We need to know the number of moving charges.

Electron current: mobile electron density wire Cross sectional area Average drift speed Electron Current

Observing magnetic field around copper wire: Can we tell whether the current consists of electrons or positive ‘holes’? In some materials current moving charges are positive: Ionic solution “Holes” in some materials (same charge as electron but +) Conventional Current The prediction of the Biot-Savart law is exactly the same in either case.

Metals: current consists of electrons Semiconductors: n-type – electrons p-type – positive holes Most effects are insensitive to the sign of mobile charges: introduce conventional current: Conventional Current Units: C/s  A (Ampere) André Marie Ampère ( )

Superposition principle is valid The Biot-Savart law for a short length of thin wire The Biot-Savart Law for Currents

Four-step approach: 1.Cut up the current distribution into pieces and draw  B 2.Write an expression for  B due to one piece 3.Add up the contributions of all the pieces 4.Check the result Magnetic Field of Current Distributions

Step 1: Cut up the current distribution into pieces and draw  B. Origin: center of wire Vector r: Magnitude of r: A Long Straight Wire

Step 2: Write an expression for  B due to one piece. Unit vector:  B field due to one piece: A Long Straight Wire :

need to calculate only z component A Long Straight Wire

Step 3: Add up the contribution of all the pieces. A Long Straight Wire

Special case: x<<L A Long Straight Wire What is the meaning of “x”?

Step 4: Check results direction  far away: r>>L  units:  A Long Straight Wire

Right-hand Rule for Wire Conventional Current Direction

v r B1B1 B2B2 Which of B 1 and B 3 is larger? A.B 1 is equal B 2 B.B 1 is larger than B 2 C.B 1 is smaller than B 2 Cklicker 45 