A B C D

Find currents through resistors Loop 1Loop 2Loop 3 Loop 4 I1I1 I2I2 I3I3 I5I5 I4I4 loop 1: loop 2: loop 3: nodes: Five independent equations and five unknowns Exercise: A Complicated Resistive Circuit

Chapter 21 Magnetic Force

The Biot-Savart law for a moving charge The Biot-Savart law for a short piece of wire: How does magnetic field affects other charges? Magnetic Field of a Moving Charge

Direction of the magnetic force depends on: the direction of B the direction of v of the moving charge the sign of the moving charge q – charge of the particle v – speed of the particle B – magnetic field Magnetic Force on a Moving Charge

Right Hand Rule for Magnetic Force

What is the effect on the magnitude of speed? Kinetic energy does not change Magnetic field cannot change a particle’s energy! Magnetic field cannot change a particle’s speed! Magnetic force can only change the direction of velocity but not its magnitude Effect of B on the Speed of the Charge

Single electron in uniform B: (v<<c) e/m e = C/kg Magnitude of the Magnetic Force

Confined area: deflection What if we have large (infinite) area with constant B  v Motion in a Magnetic Field

Any rotating vector: Circular Motion at any Speed  …angular speed Cyclotron Frequency

if v<<c: Alternative derivation: Circular motion: Period T: Circular Motion at Low Speed independent of v! Non-Relativistic

Position vector r: valid even for relativistic speeds Used to measure momentum in high-energy particle experiments Determining the Momentum of a Particle Circular motion

Determining e/m of an Electron

1897: m/e >1000 times smaller than H atom Joseph John Thomson ( )

Clicker Question

What if v is not perpendicular to B? Direction? Magnitude? Exercise

Which direction is electron going? Exercise: Circular Motion

Can combine electric and magnetic forces: Coulomb law and Biot-Savart law have coefficients 1/(4  0 ) and  0 /(4  ) to make the field and force equations consistent with each other The Lorentz Force

Is it possible to arrange E and B fields so that the total force on a moving charge is zero? B FBFB FEFE E What if v changes? A Velocity Selector

A negative charge is placed at rest in a magnetic field as shown below. What is the direction of the magnetic force on the charge? A.Up B.Down C.Into the page D.Out of the page E.No force at all. B

A.Up B.Down C.Into the page D.Out of the page E.No force at all. A negatively charged particle is moving horizontally to the right in a uniform magnetic field that is pointing in the same direction as the velocity. What is the direction of the magnetic force on the charge? B

A.Left B.Up C.Down D.Into the page E.Out of the page Now, another negatively charged particle is moving upward and to the right in a uniform magnetic field that points in the horizontal direction. What is the direction of the magnetic force on the charge? B

Current: many charges are moving Superposition: add up forces on individual charges Number of moving charges in short wire: Total force: I Force of a short wire: In metals: charges q are negative. Will this equation still work? Magnetic Force on a Current-carrying Wire

v B EE ???? + -  V>0 Hall Effect h When does it reach equilibrium?

v B EE >0 + -  V>0 v B EE ???? + -  V<0 Hall Effect for Opposite Charges

By measuring the Hall effect for a particular material, we can determine the sign of the moving particles that make up the current Hall Effect Edwin Herbert Hall ( )

What is the magnitude of the Hall effect in a metal? Monovalent metals: n is the same as # of atoms per m 3 Some metals: n is larger than # of atoms per m 3 Hall Effect in a Metal

Voltmeter 1 reading is POSITIVE Voltmeter 2 reading is POSITIVE Mobile charges are: A) Positive (holes) B) Negative (electrons) C) Not enough information Clicker Question