Sources of Magnetic Fields

AP Physics CChapter 28

Magnetic Field of a Moving Charge

http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html

Experiments support the following observations about the magnetic field:◦ Proportional to the quantity of charge◦ Proportional to inverse square of the distance

from the source point to the field point◦ Proportional to the sine of the angle between the

velocity vector and the position vector◦ Proportional to the speed of the moving charge

Magnetic Field of a Moving Charge

Further observations about the magnetic field:◦ B-Field is perpendicular to the plane containing the

line from the source point to the field point and the particle’s velocity

◦ E-Field lines radiate outwards◦ B-Field lines are circles with centers along the line of

v and lying in the planes perpendicular to this line◦ RHR is used to determine the direction of the B-Field◦ μo is called the permeability of free space and is

equal to 4π x 10-7 T-m/A

Magnetic Field of a Moving Charge

Magnetic Field of a Current Element

If the cross-sectional area of the current element is A, the volume of the segment is Adl. If there are n charges of q per unit volume, then the total charge of moving charges is dQ = nqAdl. The charges are moving with a drift velocity of vd.

This is known as Biot-Savart Law.

Magnetic Field of a Current Element

The vector dB is perpendicular both to dl (which points in the direction of the current) and to the unit vector

The magnitude of dB is inversely proportional to r2

The magnitude of dB is proportional to the current and to the magnitude dl of the length element dl

The magnitude of dB is proportional to sin θ, where θ is the angle between the vectors dl and

The Biot-Savart law is also valid for current consisting of charges flowing through space

Properties of the Magnetic Field Created by an Electric Current:

r

r

Magnetic Field of a Straight Conductor◦ Given a straight

conductor with length 2a carrying a current of I, find the B-field at a point on its perpendicular bisector, at a distance of x from the conductor.

Sample Problem #1

Calculate the magnitude of the magnetic field 4.0 cm from an infinitely long, straight wire carrying a current of 5.0 A.

Sample Problem #2

The diagram below is an end view of two long, straight, parallel wires perpendicular to the xy-plane, each carrying a current I but in opposite directions. Find the magnitude and direction of B at point P.

P d d d

Sample Problem #3

Case 1: Current in the same direction◦ B-field due to

current of conductor 1

◦ Force acting on conductor 2

Force Between Parallel Conductors

One Amp is that current which, when flowing in each of two straight, parallel, infinitely long wires, separated by 1m, in a vacuum, produces a force per unit of 2 ×10-

7Nm-1.

Definition of the Ampere:

What is the force if the current is in the opposite directions?

If the current in one wire is 2 A and the other current is 6 A, which is true:◦ The force on the second wire is 3 times the force on

the first wire.◦ The force on the first wire is 3 times the force on the

second wire.◦ The forces are equal and opposite to each other.

A loose spiral spring is hung from the ceiling, and a large current is sent through it. Do the coils move closer together or farther apart?

Force Between Parallel Conductors

The current in the long, straight wire is 5.00 A, and the wire lies in the plane of a rectangular loop, which carries 10.0 A. The long wire is 0.100 m from the rectangular loop, and the rectangular loop is 0.150 m x 0.450 m. Find the magnitude and direction of the net force exerted on the loop by the magnetic field created by the wire.

Sample Problem #4

Given a single loop of wire, with a radius a, use the Biot-Savart law to find the magnetic field at point P.

Magnetic Field of a Circular Loop

A wire carrying a current of 5.00 A is to be formed into a circular loop of one turn. If the required value of the magnetic field at the center is 10.0 μT, what is the required radius?

Sample Problem #4

A coil consisting of 100 circular loops with radius 0.60 m carries a current of 5.0 A.◦ Find the magnetic field at a point along the axis of

the coil, 0.80 m from the center.◦ Along the axis, at what distance from the center is

the field magnitude 1/8 as great as it is at the center?

Sample Problem #5

Provides an alternative formulation of the relation between a magnetic field and its source.

Used to find magnetic fields caused by some highly symmetric current distributions and to find current distributions corresponding to particular magnetic field configurations.

Analogous to Gauss’s law for determining electric fields of highly symmetric distributions of charge.

Ampere’s law valid for steady current and magnetic fields that do not vary with time.

Ampere’s Law

Consider a magnetic field caused by a long, straight conductor carrying a current I:

Ampere’s Law:

Ampere’s Law

2

(2 )2

o

o

o enclosed

IB

r

IB dl r

r

B dl I

A long, straight wire of radius R carries a steady current I that is uniformly distributed through the cross-section of the wire. Calculate the magnetic field a distance r from the center of the wire in the regions◦ r > R◦ r < R◦ Sketch a B vs. r graph

Sample Problem #6

A solenoid is a long wire wound in the form of a helix. An ideal solenoid is approached when the turns are closely spaced and the length is much greater than the radius of the turns. In this case, the external field is zero, and the interior field is uniform over a great volume. Find the magnetic field inside a solenoid.

Sample Problem #7

Consider a toroidal solenoid having N closely spaced turns of wire, calculate the magnetic field inside the solenoid and outside of the solenoid.

Toroidal Solenoid