Chapter 30: Magnetic Field Sources

DEFINITION:

Diamagnetic - substances with weak magnetic moments opposite to the applied magnetic field
Paramagnetic - substances with weak magnetic moments in the same directions as the applied magnetic field
Ferromagnetic - substances with magnetic moments between atoms that align to create strong magnetization, which remains after the external magnetic field is removed.

Magnetic flux through a closed surface is:

Biot - Savart Law

The magnetic field, dB at a point P due to an element of length ds that carries a current I is detailed in the following equation:

where:

r is the distance from the element to point P

vector r is the unit vector = 1

We find the total field at point P by integrating this expression over the entire current distribution 

Amperes Law:  

The law states the line integral     around any closed path is equal to permeability of free space times the enclosed current.

Gauss' Law for Magnetism states the total magnetic flux over a closed surface integral equals zero:

Magnitude of the magnetic field: where the field lines are concentric with the current carrying wire:

• B = [(uo)(I)]/[2pi(r)]

uo = permeability of free space

Where: 

 r = distance from a long straight wire carrying an electric current 

Magnitude of the field around a solenoid:

• B = [(N)(uo)(I)]/[2pi(r)]

uo = permeability of free space

Magnitude of the field around a toroid: 

• B = [(N)(uo)(I)]

uo = permeability of free space

Magnetic force per unit length between two parallel wires separated by a distance a, carrying current I1 and I2 has a magnitude of:

                                

= FB / legnth

Class Practice examples:

CP 11-06-13
BIOT-SAVART LAW 

1. Calculate the magnetic field outside am infinitely long, straight wire.

*set up in integral to find the Magnetic field (B) at point P

i) write the Biot Savart Law: 

ii) solve I(dl) x (r unit vector) = I(dl)(1)sin(a) = I(dl)sin(a)

iii) solve r^2    FIRST, SET AXIS & ORIGIN 

     r^2 = a^2 + x^2

iv) plug back everything back into the original equation

     dB = [(uoI(dl)sin(a)]/[4pi(a^2 + x^2)]

v) get dl in terms of x:

x = l 

vi) get r in terms of x

r = 1 unit vector 

vii) integrate[dB]dx from 0 -> infinity 

where uoI/4pi is a constant 

*note that the integral must be multiplied by 2 to encompass the entire infinite wire. 

2*(uoI/4pi) integral[dB = [(dx)sin(180)]/(a^2 + x^2)] from 0 -> infinity

2*(uoI/4pi) integral[dB = (dx)/(a^2 + x^2)] from 0 -> infinity

Conclusion: dl is always parallel to I, same direction!

AMPERES LAW

1. Find the magnetic field between 0 < r < R in a wire (imagine a cylindrical surface)

given: 

Current density (J) is uniform

R = radius 

infinite wire 

i) Create an Amperian loop 

• SURFACEint[B dot dl] = (uo)(I)

ii) pick an arbitrary direction for the current, I

iii) determine the symmetry of the problem to discuss and describe the magnetic field 

- If the magnetic field exists (B) it cannot change as you go around the loop. 

- At the origin, the physical situation at one point is the same at another point. 

- B is not a function of dl because due to the inherent symmetry of the universe. 

iv) solve the equation for the Amperian loop

SURFACEint[B dot dl] = (uo)(I)

[B] * SURFACEint[dlcos(theta)] = (uo)(I)

• Magnetic field (B) must be perpendicular to the current (I) and in the same plane as point P

           THEREFORE there can be no right or left component.  B is therefore tangent to to the                        current path. 

2. determine if the magnetic field is CW or CCW 

this can be accomplished using two different methods.

method 1: 

Use the RHR to determine the Current (I) path. 

Leave B as +/- 

Apply Amperes law

Solve for B (should be positive, magnitudes are always positive)

method 2:  apply amperes law 

i) select a direction for the magnetic field (CW or CCW)

• If B is in the direction of dl, the angle between +/- B and dl is 0
• If B is opposite of the direction of dl, the angle between +/- B and dl is 180
• If I is in the same direction as B, I is positive
• If I is opposite of the direction of B, I is negative.