1. Add up the contribution of all point charges at this point q1q1 r1r1 q2q2 r2r2 A 2. Travel along a path from point very far away to the location of interest and add up at each step: q1q1 q2q2 A E dl Potential at a Certain Location

A Example: E = 0 inside a charged metal sphere, but V is not! Common Pitfall Assume that the potential V at a location is defined by the electric field at this location.

A negative test charge Q = -0.6C was moved from point A to point B In a uniform electric field E=5N/C. The test charge is at rest before and after the move. The distance between A and B is 0.5m and the line connecting A and B is perpendicular to the electric field. How much work was done by the net external force while moving the test charge from A to B? A B 0.5m E = 5 N/C A. 1.5J B. 0J C.–1.5J D. 3.0J E.–3.0J

After moving the -0.6C test charge from A to B, it was then moved from B to C along the electric field line. The test charge is at rest before and after the move. The distance between B and C also is 0.5m. How much work was done by the net external force while moving the test charge from A to C? A. 1.5J B. 0J C.–1.5J D.3.0J E.-3.0J A C B 0.5m E = 5 N/C

Instead of moving the test charge from A to B then to C, it is moved from A to D and then back to C. The test charge is at rest before and after the move. How much work was done by the net external force while moving the test charge this time? A. 1.5J B. 0J C.–1.5J D. Infinitely big E.Do not know at this time. E = 5 N/C A B 0.5m C D

+1.5 J of work was done by the net external force while moving the -0.6 C test charge from B to C. The test charge is at rest before and after the move. What is the voltage difference between B and C, and at which point is the voltage larger? A.2.5 V, voltage higher at B. B.2.5 V, voltage higher at C. C.0.9 V, voltage higher at B. D.1.5 V, voltage higher at B. E.1.5 V, voltage higher at C. A E = 5 N/C B 0.5m C D

In a multiparticle system we can either consider a change in potential energy or a change in field energy (but not both); the quantities are equal. The idea of energy stored in fields is a general one: Magnetic and gravitational fields can also carry energy. The concept of energy stored in the field is very useful: - electromagnetic waves Potential Energy and Field Energy

e+e+ e-e- System Surroundings Release electron and positron – the electron (system) will gain kinetic energy Conservation of energy  surrounding energy must decrease Does the energy of the positron decrease?- No, it increases Where is the decrease of the energy in the surroundings? - Energy stored in the fields must decrease An Electron and a Positron

e+e+ e-e- System Surroundings Single charge: Dipole: (far) Energy: Energy stored in the E fields decreases as e + and e - get closer! An Electron and a Positron

e+e+ e-e- System Surroundings  (Field energy) +  K positron +  K electron = 0  (Field energy) = -2(  K electron ) Principle of conservation of energy: Alternative way: e + and e - are both in the system:  U el = -2(  K electron ) Change in potential energy for the two-particle system is the same as the change in the field energy An Electron and a Positron

Chapter 18 Magnetic Field

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 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 objects polarize and interact with these materials Points toward North pole – electric dipole does not do that Detecting Magnetic Fields Compass needle:

Imagine an electric circuit: What is the effect on the compass needle? What if we switch polarity? What if we run wire under compass? What if we change the current or there is no current in the wire? The Magnetic Effects of Currents

Experimental results: The magnitude of B depends on the amount of current 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 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 ( )

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 ( )

Nikola Tesla ( ) High tension coil demonstration

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

Question 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 Why must the field change direction above and below the dashed line?

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

What is the magnetic field created by an electron orbiting around the nucleus in the simple Bohr model of the H atom? v = 2.2  10 6 m/s r = 0.5  m v r (B Earth =2  T) Exercise

v r B1B1 B2B2 B3B3 Which is larger, B 1 or B 3 ? Which is larger, B 1 or B 2 ? Distance Dependence

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 + - -

Question An electron passing through the origin is traveling at a constant velocity in the negative y direction. What is the direction of the magnetic field at a point on the positive z axis? A)-x B)+x C)-z D)+z E)No magnetic field x z y v

What would the direction of conventional current have to be? A current-carrying wire lies on top of a compass. What is the direction of the electron current in this wire? Exercise

Electric fields: produced by charges Magnetic fields: produced by moving charges charged tape Any magnetic field? Frame of Reference

Must use the velocities of the charges as you observe them in your reference frame! There is a deep connection between electric field and magnetic fields (Einstein’s special theory of relativity) Frame of Reference

If we suddenly change the current in a wire: Magnetic field will not change instantaneously. Electron and positron collide: Produce both electric and magnetic field, these fields exist even after annihilation. Changes propagate at speed of light There is no time in Biot-Savart law: Speed of moving charges must be small Retardation

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

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

Example: copper wire (Cu) Molar mass = 64 g Density  = 9 g/cm 3 = kg/m 3 Each atom gives one electron Mass of one atom: Number of atoms in 1 m 3 : Typical Mobile Electron Density

Typical electron current in a circuit is ~ electrons/s. What is the drift speed of an electron in a 1 mm thick copper wire of circular cross section? Typical Mobile Electron Drift Speed

How much time would it take for a particular electron to move through a piece of wire 30 cm long? How can a lamp light up as soon as you turn it on? Typical Mobile Electron Drift Speed

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 ( )

A typical electron current in a circuit is electrons/s. What is the conventional current? Exercise

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

Moving charge produces a curly magnetic field B units: T (Tesla) = kg s -2 A -1 Single Charge: Biot-Savart Law The Biot-Savart law for a short length of thin wire Current: