Example: A single square loop of wire is placed in the plane of a magnetic field with a strength of 200 mT directed to the right. The loop has sides 10 cm long, with a current of 5 A flowing in a clockwise direction.

a) What is the magnetic moment of this loop?b) What is the torque on this loop?c) What would the torque be if the single loop was replaced by a coil of 10 loops?d) How much energy would you store in the magnetic field by rotating the loop from an

angle of 30o to an angle of 60o relative to the direction of the magnetic field?

B

x II

d

A

a)

IA 20.05Am Down

b) B sinB

10.01Nm CCW

c)

1 2 3 10 N B

0.1Nm CCW

d)U B

cosB

60 30U U U cos60 cos30B 33.66 10 J

The Cyclotron is one of the first particle accelerators.

A uniform external magnetic field moves the charges in a circular path.

An electric field accelerates the charges while they are in the gap, causing the ever increasing radius of the circular path until the charged particles are shot out of the cyclotron.

The mass spectrometer is a commonly used chemical analysis instrument.

Moving charges enter a uniform magnetic field, where ions are separated according to their charge to mass ratio.

A gas (can also use a vaporized liquid, or vaporized solid) is ionized by applying heat from a current carrying filament.

If the amount of ionization is controlled the mass of the ion and hence the elemental composition of the gas can be determined.

CH 28: Magnetic Fields of moving charges

We know that magnetic fields are created through the motion of charge. We are now going to look at one of several methods for determining the strength of the magnetic field created by a current carrying wire.

02

ˆ

4

Ids rdB

r

Biot – Savart Law

dB – small part of entire magnetic field [T]I – Current through wire [A]ds – small displacement along segment of wire [m]r – distance along radial direction from center of wire [m]m0 – permeability of free space [Tm/A]m0 = 4p x 10-7 Tm/A

• dB is perpendicular to ds (in direction of I) and to unit vector r• dB is proportional to 1/r2

• dB is proportional to I• dB is proportional to sin q, where q is the angle between ds and r

I

ds

Bout

Bin

r̂

r̂

P

P

Point P is a point a distance r from the center of the wire in the direction r.

Point P’ is a point a distance r from the center of the wire in the direction r’.

Example: What is the magnetic field strength at a distance a from a long straight wire.

I

P

a 02

ˆ

4

Ids rdB

r

x

y

02

ˆ

4

Idx ydB

r

r

ds

x

qr̂

cscsin

ar a

tana

x

cottan

ax a

2cscdx a d

o

02

sin ˆ4

IdxdB k

r

Only has contributions from the y-component. (x-component is parallel to ds)

2

02 2

csc sin ˆ4 csc

a dIdB k

a

0 ˆsin4

IdB d k

a

900

0

ˆsin4

IB d k

a

Angle when x = 0

Angle when

9000ˆcos

4

Ik

a

0 ˆcos 90 cos 04

IB k

a

0 10 ˆ4

Ika

0 ˆ2

IB k

a

Symmetric about y-

axis, so multiply by 2 for total magnetic field.

x