PES 1120 Spring 2014, Spendier
Lecture 29/Page 1
Today:
- Magnetic Field of a Moving Charged Particle
- Biot-Savart Law: magnetic field created by a current-carrying wire: Line
and loop
Thus far we have talked about magnetic fields being produced by a permanent magnet.

We assumed that there was a uniform B - field (coming from somewhere) and looked at

what would happen to moving charges we put inside B . Now we will actually calculate

B
DEMO: (next time due to missing power supply)
1819: Hans Christian Oersted found that a compass needle was deflected by a currentcarrying wire.
Direction for the Straight Wire
The magnetic field lines make concentric circles around the wire
RHR for straight wire’s magnetic
field: Point thumb in direction of the
current, your fingers curl in the
direction of the MAGNETIC FIELD
LINES.
PES 1120 Spring 2014, Spendier
Lecture 29/Page 2
Magnetic Field of a Moving Charged Particle
Why do moving charged particles experience a force by a magnetic field?
They must be creating their own magnetic field!
To find the magnetic field at point P:
 q v sin 
Experiments give us: B  0
4
r2
Permeability of Free Space: 0  4 107 T  m / A

 
Experiment also finds the B is in the direction of v  r
 
0 q v sin  0 q vr sin  0 q v  r
B


4
r2
4
r3
4 r 3
  q v  r 0 q v  rˆ
B 0

4 r 3
4 r 2
There is a magnetic field caused by a moving charge that is perpendicular to both the
direction of motion and the vector which points from the charge to the point that you
want to determine the field. The strength of the field (its magnitude) falls as the inverse
distance squared from the charge, and is proportional to the velocity.
Direction of the Field
The magnetic field of a moving charge looks very different from that of a bar
magnet! There are no poles. The field lines make circles around the charge.
RHR for moving charged particle’s magnetic field: Point thumb in direction
of the velocity, your fingers curl in the direction of the MAGNETIC FIELD
LINES created by a positive charge.
PES 1120 Spring 2014, Spendier
Lecture 29/Page 3
Example:
A pair of point charges, q = +75.0 μC and q'= −50.0 μC, are moving with speeds v = 3.00
× 105m/s and v'= 6.50 × 105m/s. When the charges are at the locations shown in the
figure, what are the magnitude and direction of
(a) the magnetic field produced at the origin and
(b) the magnetic force that q' exerts on q?
PES 1120 Spring 2014, Spendier
Lecture 29/Page 4
PES 1120 Spring 2014, Spendier
Lecture 29/Page 5
Biot-Savart Law:
To find the magnetic field created by a current-carrying wire, we modify the single
charge equation.
s
Each point on the wire contributes an infinitesimal amount to the total magnetic
field at the point P.
 0 dq v  r
dB 
4
r3


ds dq 

dqv  dq

ds  Ids
dt dt
 0 I ds  r 0 I ds  rˆ
dB 

4
r3
4
r2


0 I ds  rˆ
Law of Biot and Savart: B  
4
r2
Jean-Baptiste Biot and Felix Savart performed experiments to determine what this
magnetic field depended on and derived the above equation.
This is very similar from electrostatics:

Coulomb's Law: can add up dE from each piece of charge dq.

1 dq
dE 
rˆ
4 0 r 2
PES 1120 Spring 2014, Spendier
Lecture 29/Page 6
Magnetic field from a long (infinite) straight current carrying wire
We want to find the magnetic field from an infinite line of current at a point x away from
the line along a perpendicular sector.
PES 1120 Spring 2014, Spendier
Magnetic Field from a Current Loop
Lecture 29/Page 7