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1
MAGNETISM

A magnetic field is a vector field that permeates space and which can exert a magnetic force on moving
electric charges and on magnetic dipoles.
We define the magnitude of the magnetic field by measuring the magnetic force on a moving charge q:
Fmag
N∙s
B=
1 T(Tesla) =
|q|v
C∙m
Direction at any location is the direction in which the north pole of the compass needle points at that location.
Outside magnet: N → S
Inside magnet: S → N
(always closed loops)
1. An electric charge experiences a magnetic force when moving in a magnetic field.
Magnetic force acting on a charge q
Magnetic force on a wire carrying current I
in a magnetic Field B: F = qvB sinin a magnetic field B: F = I LB sin
 

q = charge [C]
v = velocity [m/s]
B = magnetic field [Tesla T]
 = angle between v and B
I = current [A]
L = length [m]
B = magnetic field [T]
 = angle between I and B
R-H-R 1: The direction of the magnetic force on a charge/current is given by the right-hand rule 1:
Outstretch fingers in the direction of v (or current I).
Curl fingers as if rotating vector v (I ) into vector B.
Magnetic force on a positive charge (or I) is in
the direction of the thumb.
Magnetic force on a negative charge points in opposite direction.

Charge q in elec. field E and mag. field B
The electric force: Felec = Eq
The magnetic force: Fmag = qvB sin
● is always parallel to the direction of the
● is always perpendicular
electric field.
● acts on a charged particle
independent of the
particle’s velocity (even at rest).
● does the work
when moving charge.
The work, W = Fel d cos θ1, is converted into kinetic
energy which is, in the case of conductors,
transferred to thermal energy through collisions
with the lattice ions causing increased amplitude
of vibrations seen as temperature rise.
to the direction of the
magnetic field
● acts on a charged particle only when the particle is in motion
and only if v and B do not point in the same or opposite
direction (sin 00 = sin 1800 = 0).
●
Force is perpendicular to the direction of the motion,
so the work done by magnetic force is zero.
W = Fmagd cos θ1 = 0 (cos 900 = 0).
W = ΔKE = 0
Hence change in kinetic energy of the charge is 0,
and that means that mag. force cannot change
the speed of the charge. Magnetic force can only
change direction of the velocity – therefore it acts as
centripetal force.
θ1 is angle between F and direction of motion (v and d)
2
Examples of the Lorentz Force
Two important applications of the Lorentz force are
1) the trajectory of a charged particle in a uniform magnetic field and
2) the force on a current-carrying conductor.
1) The trajectory of a charge q in a uniform magnetic field B
Force is perpendicular to B,v
B does no work! (W = F d cos θ1 )
● Speed is constant (W = Δ KE = 0 )
● Circular motion
●
●
Centripetal force definition:
Motion in a circle represents accelerated motion, and requires a force directed toward the center of the circle.
This force is called the centripetal force which means "center seeking" force.
It is forcing mass m to move in the circle of radius R with the speed v .
The force has the magnitude
𝐹𝑐 = 𝑚
𝑣2
𝑅
= 𝑚 𝑎𝑐
In the case the charge q is subject to the uniform field B, centripetal force Fc
is magnetic force forcing the charge to move in a circle:
Positive charge q in
magnetic field B
𝐹𝑐 = 𝐹𝑚𝑎𝑔
𝑚
𝑣2
𝑅
𝑹=
●
●
= 𝑞𝑣𝐵
𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑝𝑖𝑐𝑡𝑢𝑟𝑒 𝜃 𝑖𝑠 900 → sin 900 = 1
𝒎𝒗
𝒒𝑩
massive or fast charges – large circles
large charges and/or large B – small circles
B is represented by the crosses – into the page
R is the radius of the path
F is magnetic force on the charge directed
toward the centre of the circular path
2. A moving charge produces a magnetic field.
R-H-R 2: The direction of the magnetic field produced by electric current is given by the right-hand rule 2:
If a wire is grasped in the right hand with the thumb in the direction of current flow, the fingers will curl in the
direction of the magnetic field.
3
Magnetic field B around a wire with current I
B=
𝜇0 𝐼
2𝜋 𝑟
= the permeability of free space 4×10-7 T·m/A
I = current [A]
r = distance from the center of the conductor
n = N/L number of turns of wire per unit length
Magnetic Field B Inside of a Solenoid
B = 0 n I
The magnetic field is concentrated
into a nearly uniform field in the
centre of a long solenoid. The field
outside is weak and diverging
Magnetic Force per unit length between Two Parallel Wires:
F1 F2  0 I1 I 2


L
L 2 d
(a) two parallel currents – the force between them is attractive
Current I1 creates magnetic field B1
at distance d from the wire
B1 =
Place a wire with current I2
at distance d from I1
𝜇0 I1
The force on I2 is:
2𝜋 𝑑
𝜇0 I1
2𝜋 𝑑
F2 = L I2 B1 = L I2
F2 = L
𝜇0 I1 I2
2𝜋 𝑑
(b) two antiparallel currents – the force between them is attractive
Current I1 creates magnetic field B1
at distance d from the wire
B1 =
𝜇0 I1
Place a wire with current I2
at distance d from I1
The force on I2 is:
2𝜋 𝑑
F2 = L I2 B1 = L I2
F2 = L
𝜇0 I1 I2
2𝜋 𝑑
𝜇0 I1
2𝜋 𝑑
The forces exerted on the wire #1 and wire #2 are of the same magnitude but in opposite direction
(consistent with Newton’s third law)
One Ampere is defined as that current flowing in each of two infinitely-long parallel wires of negligible
cross-sectional area separated by a distance of one metre in a vacuum that results in a force
of exactly 2 x 10-7 N per metre of length of each wire.