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 current-carrying 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:

Experiments give us: 02

sin4

q vB

r

Permeability of Free Space: 70 4 10 /T m A

Experiment also finds the B

is in the direction of v r

0 0 02 3 3

sin sin4 4 4

q v q vr q v rBr r r

0 03 2

ˆ4 4

q v r q v rBr r

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.

Each point on the wire contributes an infinitesimal amount to the total magnetic field at the point P.

034

dq v rdBr

ds dqdqv dq ds Idsdt dt

0 03 2

ˆ4 4

I ds r I ds rdBr r

Law of Biot and Savart: 02

ˆ4

I ds rBr

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.

20

1 ˆ4

dqdE rr

s

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 Lecture 29/Page 7 Magnetic Field from a Current Loop