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Part 3: Magnetic Force
3.1: Magnetic Force & Field
A. Charged Particles
A moving charged particle creates a magnetic field vector at every point in space except at its
position.
Symbol for Magnetic Field
B
mks units
[Tesla = T = N/(Am)]
other common unit
[Gauss = G] [1 G = 10-4 T]
This magnetic field exerts a magnetic force on another moving charged particle.
B
Size:
q

v
q = charge of particle
v = speed of particle
 = angle between magnetic field & velocity vectors
Direction: Force Right Hand Rule #1 (Force RHR #1)
1. Curl fingers from v direction into B direction.
2. Thumb gives force direction for + charge. Force direction is opposite for – charge.
The size and direction of the magnetic force can be expressed with one vector equation using the
cross product.
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B. Motion of Charged Particles in B-Fields
If a charged particle moves through a uniform magnetic field, the path of the particle will be:



straight (no force) if the angle between the velocity and field is 0 or 180°
a circular arc if the angle between the velocity and field is 90°
a spiral if the angle between the velocity and field is some other angle
Circular Arc Motion
For the circular arc, where the velocity is perpendicular to the field, the force is a centripetal
force which gives the particle a centripetal acceleration.
m = mass of particle
r = radius of circle
This force equation can be solved for the relevant quantity.
-----------------------------------------------------------Application 1: Synchrotron (Particle Accelerator)
Accelerate charge particles with an electric force as they travel a circular
path with a fixed radius r. The magnetic field necessary to keep them
moving in that circle is found from the above force equation.
------------------------------------------------------------
r
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Application 2: Mass Spectrometer
Ionize atoms/molecules. Accelerate ions using an electric field. Send ions into a velocity
selector so that the ions that make it through the selector have the same speed. Send these ions
into a magnetic field so that they travel circular arcs. Different masses give different arc radii so
the masses of the ions can be distinguished. Thus, the ion species can be determined.
v
ions
B
B
E
accelerating
plates
velocity
selector
(perpendicular
E & B fields)
circular arc
region
(B field only)
In the velocity selector, the electric force balances the magnetic force only for the ions that
travel at the target speed of
Inside the arc region, the mass of the ion is found from the above force equation.
C. Straight Wire
A current-carrying straight wire segment sitting in a magnetic field feels a magnetic force.
Size:
I = current
= length of segment
 = angle between field & current direction
Direction: Force Right Hand Rule #2 (Force RHR #2)
1. Curl fingers from I direction into B direction.
2. Thumb gives force direction.
B

I

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The size and direction of the magnetic force can be expressed with one vector equation using the
cross product.
D. Current Loops & Motors
If a current-carrying coil of wire is sitting in a magnetic field, the net magnetic force on the loop
is zero. However, there can be a net torque on the loop that can cause it to rotate about a fixed
axis.
Peak torque [N-m]
Average torque [N-m]
Mechanical power (due to rotation) [W = J/s]
N = number of turns in coil
B = magnetic field [T]
A = area of one turn [m2]
I = current through coil [A]
________________________________
Example:
An electric motor has 200 windings where each winding has an area of 20 cm2. The coil sits in a
magnetic field of 0.01 T. A current of 5 A flows through the coil when it rotates at 600 rpm.
Find (a) the average torque exerted on the coil and (b) the average power it delivers.
Ans. (a) 0.127 N-m (b) 8 W
____________________________________________
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3.2: Sources of Magnetic Field
Biot-Savart Law
A moving charge produces a magnetic field at a point P given by
direction
of I
P
x
r
ds
+ r
A. Straight Wire
Field at a point due to the current in a straight wire segment
P
Size:
x
1
a
2
I
Direction: Field Right Hand Rule #1 (Field RHR #1)
1. Place thumb along current I so that fingers touch point P.
2. Curl fingers towards palm. The initial direction that the fingers move is B direction.
Special Case: Long wire (10 and 2180)
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B. Two Parallel Wires
If currents flow through each wire segment, then these currents produce magnetic fields. These
fields exert magnetic forces on the wires, equal and opposite in size.

Size of force:
I1
I2
F1
F2
C. Current Loop
Field on the central axis of a loop of radius R with a current I
I
Size:
R
x
Direction: Field Right Hand Rule #2 (Field RHR #2)
1. Follow I around loop with fingers.
2. Thumb gives direction of B along axis.
Special Case: At loop’s center (x=0)
x
P
a
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D. Solenoid (Coil)
Field at a point on the central axis of a solenoid with length  and N turns
N turns
Size:
1 2
R
x
I
P

Direction: Field Right Hand Rule #2 (Field RHR #2)
1. Follow I around coil with fingers.
2. Thumb gives direction of B along axis.
Special Case: Long solenoid (190 and 290)
E. Gauss’ Law for Magnetism & Ampere’s Law
Gauss’s Law for Magnetism states that the magnetic flux through any closed surface is zero.
This is a statement of the fact that magnetic field lines form closed loops. There are no magnetic
monopoles observed in nature.
Ampere’s Law is an alternative to the Biot-Savart Law for finding the magnetic field due to a
current. It is sometimes easier to use than the Biot-Savart Law.
Like the Biot-Savart Law, Ampere’s Law states that the magnetic field strength is proportional to
the current.