PHYSICS

of electric charge; likewise, a wire with current passing through it
creates a magnetic field due to the charge moving in the wire. Bar
magnets also produce a magnetic field.
Let us define a magnetic field vector lJ (sometimes called the
magnetic induction) at some point in space in terms of a magnetic
force would be exerted on an appropriate object - a charged
particle moving with a velocity V. Magnetic force (Lorenz force)
acting on a particle with charge q depends on the velocity V of
the charge and is equal to
PM = q [V x B] (18.1)
where [V x Bj is the cross product of vectors V and lJ (fig. 18.1).
This force is always perpendicular to both vectors V and s.
Fig. 18.1. The Lorenz force PM
acting on a charged particle with
charge q moving with a velocity
V in the presence of a magnetic
field lJ
_
F
M
F M
i3
_v
Fig. 18.2. The Lorenz force: a ­
the directions of the magnetic
force PM' velocity V of charged
particle, and magnetic field s,
b - the right-hand rule for the
determining the direction of the
The right-hand rule for the determination the direction of the
Lorenz force (fig. 18.2): you point the four fingers of your right
hand along the direction of V, and then turn them until they point
along the direction of lJ,. the thumb then points in the direction
of PM'
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a
b
",It
;~~;
(~
~
:J~
'I,
The unit of the magnetic field is the tesla (1 T = 1 N/Am).
For a large magnetic field, the commonly used subunit is the
gauss (I T = 10 4 G), e.g., the magnetic field of the Earth is
about 0.5 G, while the magnetic field of a large electromagnet is
about 1 T.
Example. A proton moves with a speed of 8.10 6 rn/s along the
x axis. It enters a region where there is a field of magnitude 2.5 T,
directed at an angle of 60° to the x axis. Calculate the initial magnetic
force and acceleration of the proton.
Solution. From Equation (18.1) , we get:
F = q Vbsine = (1.6.10- 19 C)(8·1 0 6 m/s)(2,5 T)(sin60o) = 2.77-10- 12 N.
18.2. THE MASS SPECTROMETRY
Consider the case of positively charged particle moving in a
uniform external magnetic field with its initial velocity vector
perpendicular to the field. Let us assume that the magnetic field
is into the page. Charged particle moves in a circle whose plane is
perpendicular to the magnetic field. As the force PM deflects the
particle, the directions of V and PM change continuously. Therefore
the force PM is a centripetal force. From Newton's second
law, we find that
or
F = qVB = mV 2
r = mV
qB
r
(18.2)
(18.3)
The mass spectrometer is an instrument that separates atomic
and molecular ions according to their mass-to-charge ratio. A
beam of ions first passes through a velocity selector and then
enters a uniform magnetic field (f3) directed into the paper
(fig. 18.3). Upon entering the magnetic field, the ions move in a
semicircle of radius r before striking a photograph plate at P.
From Equation (18.3) we can express the ratio m/o as
m
·I q
rB
V
(18.4)
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