Cyclotron, finalThe cyclotron’s operation is based on the fact that T is independent of the speed of the particles and of the radius of their path

When the energy of the ions in a cyclotron exceeds about 20 MeV, relativistic effects come into play

2 2 221

2 2q B RK mv

m= =

Cyclotron, 2D1 and D2 are called dees because of their shapeA high frequency alternating potential is applied to the deesA uniform magnetic field is perpendicular to them

Hall EffectWhen a current carrying conductor is placed in a magnetic field, a potential difference is generated in a direction perpendicular to both the current and the magnetic fieldThis phenomena is known as the Hall effectIt arises from the deflection of charge carriers to one side of the conductor as a result of the magnetic forces they experience

Hall VoltageThis shows an arrangement for observing the Hall effectThe Hall voltage is measured between points a and c

Hall Voltage, finalΔVH = EHd = vd B d

d is the width of the conductorvd is the drift velocityIf B and d are known, vd can be found

RH = 1 / nq is called the Hall coefficientA properly calibrated conductor can be used to measure the magnitude of an unknown magnetic field

HH

I IB R BVnqt t

Δ = =

Quick Quiz 29.4

The four wires shown below all carry the same current from point A to point B through the same magnetic field. In all four parts of the figure, the points A and B are 10 cm apart. Which of the following ranks wires according to the magnitude of the magnetic force exerted on them, from greatest to least? (a) b, c, d (b) a, c, b (c) d, c, b (d) c, a, b (e) No force is exerted on any of the wires.

Answer: (a). (a), (b) = (c), (d). The magnitude of the force depends on the value of sin θ. The maximum force occurs when the wire is perpendicular to the field (a), and there is zero force when the wire is parallel (d). Choices (b) and (c) represent the same force because Case 1 tells us that a straight wire between A and B will have the same force on it as the curved wire.

Quick Quiz 29.4

Quick Quiz 29.6

Rank the magnitudes of the torques acting on the rectangular loops shown in the figure below, from highest to lowest. (All the loops are identical and carry the same current.) (a) a, b, c (b) b, c, a (c) c, b, a (d) a, c, b. (e) All loops experience zero torque.

Quick Quiz 29.7

Rank the magnitudes of the net forces acting on the rectangular loops shown in this figure, from highest to lowest. (All the loops are identical and carry the same current.) (a) a, b, c (b) b, c, a (c) c, b, a (d) b, a, c (e) All loops experience zero net force.

Quick Quiz 29.10a

Three types of particles enter a mass spectrometer like the one shown in your book as Figure 29.24. The figure below shows where the particles strike the detector array. Rank the particles that arrive at a, b, and c by speed.(a) a, b, c (b) b, c, a (c) c, b, a (d) All their speeds are equal.

Answer: (d). The velocity selector ensures that all three types of particles have the same speed.

Quick Quiz 29.10a

Quick Quiz 29.10b

Rank the particles that arrive at a, b, and c by m/q ratio.(a) a, b, c (b) b, c, a (c) c, b, a (d) All their m/q ratios are equal.

Chapter 30

Sources of the Magnetic Field

Biot-Savart Law – IntroductionBiot and Savart conducted experiments on the force exerted by an electric current on a nearby magnetThey arrived at a mathematical expression that gives the magnetic field at some point in space due to a current

Biot-Savart Law – Set-UpThe magnetic field is dB at some point PThe length element is dsThe wire is carrying a steady current of I

Biot-Savart Law –Observations

The vector dB is perpendicular to both ds and to the unit vector directed from ds toward PThe magnitude of dB is inversely proportional to r2, where r is the distance from ds to P

r̂

Biot-Savart Law –Observations, cont

The magnitude of dB is proportional to the current and to the magnitude ds of the length element dsThe magnitude of dB is proportional to sin θ, where θ is the angle between the vectors ds and r̂

The observations are summarized in the mathematical equation called the Biot-Savart law:

The magnetic field described by the law is the field due to the current-carrying conductor

Biot-Savart Law – Equation

24ˆIoμ dd

π r×

=s rB

Permeability of Free SpaceThe constant μo is called the permeability of free spaceμo = 4π x 10-7 T. m / A

Total Magnetic FielddB is the field created by the current in the length segment dsTo find the total field, sum up the contributions from all the current elements Ids

The integral is over the entire current distribution

24ˆIoμ d

π r×

= ∫s rB

B for a Long, Straight ConductorThe thin, straight wire is carrying a constant current

Integrating over all the current elements gives

( )

2

1

1 2

4

4

I sin

I cos cos

θo

θ

o

μB θ dθπaμ θ θπa

=

= −

∫

( ) sin ˆˆd dx θ× =s r k

B for a Long, Straight Conductor, Special Case

If the conductor is an infinitely long, straight wire, θ1 = 0 and θ2 = πThe field becomes

2IoμB

πa=

B for a Long, Straight Conductor, Direction

The magnetic field lines are circles concentric with the wireThe field lines lie in planes perpendicular to to wireThe magnitude of B is constant on any circle of radius aThe right-hand rule for determining the direction of B is shown

B for a Circular Current LoopThe loop has a radius of R and carries a steady current of IFind B at point P

( )2

32 2 22

Iox

μ RBx R

=+