Magnetic Field and Work

Magnetic Field and Work

?Magnetic force is always perpendicular to velocity

TWhheyr?efBoerecaBusfieeldKdoe=sFrno

wxror=k!Fr

(

vrt

)

=

0

?Consequences

Kinetic energy does not change

Speed does not change

Only direction changes Particle moves in a circle

(if

vr

r B

)

PHY2049: Chapter 28

1

Trajectory in a Constant Magnetic Field

?A charge q enters B field with velocity v perpendicular to B. What path will q follow?

Force is always velocity and B

Path will be a circle. F is the centripetal force needed to keep the charge in its circular orbit. Let's calculate radius R

x x x x x x x x x x x x xBx

x x x x x x x x x x x x x x

x x x x x x x x x x x x x x

x x x x x x x x x x x x vx x

vF

F

R

PHY2049: Chapter 28

v Fq

2

Circular Motion of Positive Particle

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x xB

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x

mv2 = qvB R

R = mv qB

PHY2049: Chapter 28

v Fq

3

Magnetic Force

?Two particles of the same charge enter a magnetic field with the same speed. Which one has the bigger mass?

A

B Both masses are equal Cannot tell without more info

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x

R = mv qB

Bigger mass means larger inertia, less acceleration, thus bigger radius

x x x x x x x x x x x x x x x x x x x x x x x x

AB

PHY2049: Chapter 28

4

Work and Energy

?A charged particle enters a uniform magnetic field. What happens to the kinetic energy of the particle?

(1) it increases (2) it decreases (3) it stays the same (4) it depends on the direction of the velocity (5) it depends on the direction of the magnetic field

Magnetic field does no work, so K is constant

PHY2049: Chapter 28

5

Cosmic Ray Example

?Protons with energy 1 MeV move earth B field of 0.5 gauss or 5 ? 10-5 T. Find radius & frequency of orbit.

K

=

1 2

mv2

v=

2K m

( )( ) K = 106 1.6?10-19 =1.6?10-13 J

m = 1.67 ?10-27 kg

R = mv = 2mK eB eB

R = 2900 m

f

=1 T

=

v

2 R

=

v

2 (mv / eB)

=

eB

2 m

f = 760 Hz

Frequency is independent of v!

PHY2049: Chapter 28

6

Helical Motion in B Field

?Velvor c=itvyr

o+fvrpa(rptiacrlaellehlatos

2 components

B and perp. to B)

Only v = v sin contributes to circular motion

v||

v|| = v cos is unchanged

?So the particle moves in a helical path v

v

v|| is the constant velocity along the B field v is the velocity around the circle

B

R = mv qB

PHY2049: Chapter 28

7

Helical Motion in Earth's B Field

Particles moving along field lines cause Aurora Borealis and Australis:

PHY2049: Chapter 28

8

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