AP C - UNIT 9Magnetic Fields and Forces

Magnetic Materials

• All magnetic materials have two poles – Labeled: North and

South poles– Like poles repel

each other and opposite poles attract.

Magnetic Monopoles

Unlike electrostatics, magnetic monopoles have never been detected.

Modern View of Magnetism

Magnetism is associated with charges in motion charges in motion (currents):(currents):

–microscopic currents in the atoms microscopic currents in the atoms of of magnetic materialsmagnetic materials..

–macroscopic currents in the macroscopic currents in the windings of an windings of an electromagnetelectromagnet..

Magnetic Field

Similar to E-field, but where attraction and repulsion is occurring due to magnetism, either magnets or moving charges.

Magnetic Field Lines

The Earth’s Magnetic Field

The spinning iron core of the earth produces a magnetic field.

Magnetic Fields

Magnetic field is a vector. It has direction and can be cancelled by another field.

Magnetic Field UnitsThe SI unit of magnetic field is the Tesla (T).Dimensional analysis:

1 T = 1 N·s/(C·m) = 1 V·s /m2

Sometimes we use a unit called a Gauss (G):1 T = 104 G

The earth’s magnetic field is about 0.5 G.

Magnetic Force on Moving Charges

A charged particle in a static magnetic field will experience a magnetic force only if the particle is particle is movingmoving.

If a charge q with velocity v moves in a magnetic field B and v makes an angle w.r.t. B, then the magnitude of the force on the charge is:

Finding the Direction of Magnetic Force

• The direction of the magnetic force is always perpendicular to both and

• Fmax occurs when is perpendicular to

• F = 0 when is parallel to

v

B

v

B

v

B

Notation

To represent the z-axis on an xy plane of paper.

• • •• • •

B out of the page

B into the page

Right Hand Rule• Draw vectors v and B with

their tails at the location of the charge q.

• Point fingers of right hand along velocity vector v.

• Curl fingers towards Magnetic field vector B.

• Thumb points in direction of magnetic force F on q, perpendicular to both v and B.

• Use RHR to find F on moving charged particle for each situation a through f

Motion of Charges in B FieldsIf a charged particle is moving perpendicular to a uniform magnetic field, its trajectory will be a circle because the force F=qvB is always perpendicular to the motion, and therefore centripetal.

If path of charge is not perpendicular to field where it enters field at some angle, charge will follow a spiral path (helix) which will spiral around the B-field.

The component of velocity that is parallel to the magnetic field is unaffected. Its circlular motion will drift at a constant speed along the magnetic field

The perp component of the velocity to B-field causes the particle to

executes uniform circular motion perpendicular to the magnetic field.

Velocity SelectorA particle is accelerated through a potential difference where it then enters a magnetic and electric field. If the forces are balanced, the particle will move only horizontally between the plates (assuming negligible gravity)

Mass SpectrometerThe mass spectrometer utilizes the velocity selector and is an instrument which can measure the masses and relative concentrations of atoms and molecules. The radius of turn yields the mass of the particles.

An ion is accelerated through a voltage of 1000V with a charge of 1.6x10-19C after which it enters a chamber with a uniform B-field equal to 80mT. The ion follows a semi-circle path and strikes a photographic plate 1.62m from where it entered. Find its mass.

Hall EffectIf an electric current flows through a conductor in a magnetic field, the magnetic field exerts a force on the moving charge carriers which tends to push them to one side of the conductor. A buildup of charge at the sides of the conductor will balance this magnetic influence w/ E-field, producing a measurable voltage between the two sides of the conductor. The charge carries will then just pass through in a straight line.

Force on a WireSimilar to the force on a moving charge in a B field, we have for a conductor of length L carrying a conventional current of I in a B field. The force experienced by the conductor is:

A thin, massless, uniform rod with length 0.20m is attached to a frictionless hinge at point ‘P’. A horizontal spring (k = 4.8 N/m) connects the other end of the rod to a vertical wall. A uniform B-field equal to 0.34T is shown and a 6.5A current exists in the rod directed towards the hinge. How much energy is stored in the spring?

B

‘P’

I

53.1o

FS

FM

Torque on a current loopA wire loop is freely pivoting in a uniform B-field (+x) as shown. Each side is length ‘a’.

a

a

B

Electric Motor• An electric motor converts

electrical energy to mechanical energy– The mechanical energy is

in the form of rotational kinetic energy

• When area vector of loop is parallel to field there is no torque

After the loop moves a ½ turn, the current needs to switch direction to keep it rotating. Inertia will carry it past the edge (top of rotation) and at that moment if the current is switched, the loop keeps going. If the current isn’t switched and it passes the edge, it will rotate the other way and get nowhere

Biot Savart LawThis law is seen as the magnetic equivalent of Coulomb's Law (brute force way of doing it vs Gauss) . This finds B at a point P, a distance r from the differential element of current Ids.

0 = 4 10-7 Tm/A, magnetic permeability of free space

Current makes B-fields

Biot-Savart Law – Set-UpdB

ds

The magnetic field is at some point P

The length element is

The wire is carrying a steady current of I

• is the field created by the currentin the length segment ds• To find the total field, sum up thecontributions from all the currentelements I

– The integral is over the entire current distribution

dB

ds

B-Field due to a long straight wire along y-axis

0

ds

Example

b

a

Consider wire bent into the shape shown above. Find B at ‘P’ if current is flowing clockwise in wire.

P

I

Magnetic Field from a Wire, RHR #3The magnetic field lines from a current form circles around a straight wire with the direction given by another right hand rule.

The magnitude of the magnetic field a distance r from a straight wire is given by (just proven with rigorous proof)

Force between 2 current-carrying wiresWhat happens when current as shown in 2 parallel wires?

I1

I2

a

Force between two parallel wires

a

Find BNET at point P

P

45o

d

R

I1 I2

I2 = I1

45o

Loop of Current

Consider a coil of radius R with current flowing CW. Find B at center of coil.

X

2

ˆ

4 r

rsIddB o

A loop of Current is RHR #4

Fingers are current direction and thumb is magnetic north pole

I

d

I

A current I flows in the positive y direction in an infinite wire; a current I also flows in the loop as shown in the diagram.

What is the direction of the net force on the loop?

(a) F = -x (b) F = 0 (c) F = +x

(d) F = +y (e) F = -y

B-field due to a current loop a distance ‘x’ away

frontal view of loop of current

ds

Ampere’s LawAmpere’s Law is to magnetic fields as Gauss’ Law was to electric fields. Both are used for high symmetry problems.

Similar to drawing a Gaussian surface

NOW apply Ampere’s Law to find B surrounding long straight current carrying wire (already did this using Biot Savart)

Consider long wire with current I into page.

sdr

Magnetic field inside a wireFind B inside the wire with uniform current a distance r from the center where r < R

B-field of a solenoid Solenoid is a series of loops of current. One end acts like a N & the other like a S-pole. Ideal solenoid is approached when turns are closely spaced and length is much greater than radius of turns.

Current would flow from left to right across top of solenoid

Derivation of B inside solenoid

Cross sectional view of solenoid showing current into page on top, out of page on bottom

Derivation of B inside solenoid

3 42

1

Toroid - Solenoid shaped like a doughnut

A current I flows in an infinite straight wire in the +z direction as shown. A concentric infinite cylinder of radius R carries current 2I in the -z direction.

xx

xx

xx

x

x

2II

a

b

x

ya) What is the magnetic field Bx at point ‘a’, just outside the cylinder as shown? Take CW as positive for B.

c) Bx(b) > 0(a) Bx(b) < 0

c) Bx(a) > 0a) Bx(a) < 0 b) Bx(a) = 0

b) What is the magnetic field Bx at point ‘b’, just inside the cylinder as shown?

b) Bx(b) = 0

Two cylindrical conductors each carry current I into the page as shown. The conductor on the left is solid and has radius R=3a. The conductor on the right has a hole in the middle and carries current only between R=a and R=3a.

3a3a

aI I

What is the relation between the magnetic field at R = 6a for the two cases (L=left, R=right)?

a) BL(6a)< BR(6a) b) BL(6a)= BR(6a) c) BL(6a)> BR(6a)

Two cylindrical conductors each carry current I into the page as shown. The conductor on the left is solid and has radius R=3a. The conductor on the right has a hole in the middle and carries current only between R=a and R=3a.

3a3a

aI I

What is the relation between the magnetic field at R = 2a for the two cases (L=left, R=right)?

a) BL(2a)< BR(2a) b) BL(2a)= BR(2a) c) BL(2a)> BR(2a)

Consider a thin, infinitely large sheet of current that carries a linear current density, λ. The current is out of the page. Find B near the sheet.

Recall a similar problem with infinitely charged sheet of charge density, σ.