» Syllabus

Topic 4 (Magnetic Field, Force, & Torque

Home-Work Set for Topic 4 :

PRACTICE QUESTION SUGGESTIONS ...( NOT for grading ...) from Young 9th edition
format codes :
      bold : Basic Bread-and-butter ... you'd better know how to do `em
      Italic : Important Ideas In here ... at least read `em
      Underline : Understand the Underlying structure ... not just the answer

ch.20 conceptual quests 1,2,4,8,9,10,13,14,15 ;
      multiple-choice 1,2,3,4,5,6,7,8,9,12,13,14,15 ;
      problems 1,3,4,6,7,9,10,11,14,15,16,17,18,19,21,23,24,25,27("gauss"),29,31,32,34,35("mag.moment"),37,39 ;
      . . . 41,43,44,47,49,50,52,57,59,63,67,68,71,75,78,79,83,86,89,90,91,93

Physics II Topic 4 Summary

Idea # 8, What does Magnetic Field DO ? B deflects Moving charge - - - - - - - - - -

A "correct" compass needle will align along (parallel to) the Magnetic Field at its location.
. . . the Magnetic field is a vector . . . with strength and direction that varies from place to place (varies with time, too!)
      strength at Earth's surface is about 50 micro-Tesla [μT] in SI units . . . (½ gauss, in cgs units)
      mostly downward, but tilted about 30° Northward (horizontal at the Equator)
. . . the needle's "geographic North end" (glow-in-the-dark or red) acts as the vector arrow's tip .

As a (+) charge q moves within a magnetic field, relative to the magnet, a Force is exerted on the moving charge
. . . the Force strength is proportional to the B-field , and proportional to qv
      (...this symmetry between Source and Influence is necessary for Newton's 3rd Law to be true for Magnetic Forces ...)
. . . the Force is perpendicular to   qv , and also perpendicular to B
      (...recall that the B-field is perpendicular to its moving source   Qv ...)
=> F = qv × B   . . . use right-hand-rule #1 for this cross-product (the one for Torque): Earthlings use right-hand) .

Magnetic Force tends to make freely moving charges travel in a circle
. . .recall (Physics 1) that Force perpendicular to velocity leads to a curved path , with radial acceleration component
      a·r = −v·v = a r sin φ .   IF the magnetic Force is the only one acting,
=> r = mv/qBsinφ

Magnetic Field does zero Work on freely moving charges
. . . the Force is always   |   to the velocity (displacement is v Δt)
=> so there is no such thing as a magnetic scalar potential

Geometry & Units:   Q v = I L - - - - - - - - - - -

A line of charges moving with the same velocity as each other (parallel to the line) appears to be a current along the Line's length .

Add up the Magnetic Force on all charges moving within the object - - - - - - - - - -

Magnetic Force applied to a straight Wire, carrying Current
F = IL × B , where L is the Length of current IN the magnetic field .

Magnetic Force on a Wire Loop is zero , if the external B is uniform.

Non-uniform Field can apply net Force to a current loop
=> parallel current loops attract

Idea # 9, How To Make B? Magnetic Field Source: "Charge in Motion" is encircled by Magnetic Field B - - - - - - -

As a (+) charge Q moves away, relative to the observer, the Magnetic Field B points in a clockwise sense
. . . the B-field is stronger for more charge, and for faster motion   Qv
      think of   Qv   (one entity) as the Source of its Magnetic Field
. . . the B-field is strongest close to the moving charge   Qv
      decreasing as ¹/_r ² at large distances r from an individual source charge
. . . the B-field is strongest in a "ring" around the moving charge, 90° from its motion "axis" along the   Qv   vector .
      decreasing as sin(θ) at other orientations ... so is zero directly in front of Qv (and zero behind it).
=> B = (k_magnetism) Qv sin(θ)/r²   =   (k_magnetic) Qv × r^ /r ²
      . . . here,   r^   is the unit vector in the direction toward the "field point" (place of interest)
      . . . and the "magnetic Units constant"   k_magnetic   is usually written   ^μ/4π   (recall gravity's G, and electricity's k_c)

1) The cross product "×" uses the same right-hand-rule as was introduced in Physics 1 for torque :   r × F = τ .
. . . with right arm or wrist aligned with the r vector (extends from rotation axis to Force application point) ,
      fingers flip to align with F vector (sweeping thru angle φ ... that letter φ is spelled "phi" but pronounced "fie") ;
=> thumb points   |   to r and   |   to F , in the direction of the τ vector , with magnitude   r F sin φ

2) "encircling the piercing vector" follows the geometry of spin and angular momentum :
. . . align right thumb in the direction of the angular momentum vector, along the spin axis L = r × p
      right fingers wrap around the axis in the direction of the momentum for the individual pieces.

It is often useful to think of magnetic Field lines, which form closed loops around the Source charge as it moves
      with arrows to show the B-field direction
. . . in this case, right hand grabs the moving charge with thumb along Qv , and fingers wrap along B lines .

Add up the Magnetic Field contributions from all nearby moving charges - - - - - -

Current flowing along a Long, Narrow, Straight wire

A straight line of charge moving along its length is encircled by B (clockwise as seen from behind them)
. . . they appear as a current ... you can sum (or integrate) the the Biot-Savart formula for them all , using Qv/L = I δx .
      for an infinitely-long Current Line , the "effective Length" is only 2d , for a location at distance d from the Line
      ... the charges farther than that are such distant sources (1/r squared!) with v at such shallow angle (sin φ) so they don't contribute much .
. . . when you plug this effective length into the Biot-Savart formula (previous section), the 2 cancels and the d cancels (with one of the "r" on the bottom)
=> B_{∞ line} = μ I /2π d  
      . . . if the wire really is only 2 d long , the magnetic field is .707 (=sin 45°) this intense; 4 d long produces 90% =sin(Arctan(2))
      same amount of "magnetic flux" pierces into page below Il (for Il to right) , as pierces out from the page, above it.

If the Current is Spread Out (  |   to Qv) , use Ampere's Law

Ampere dissects the formula just derived. At distance d from the Current Line, there's a magnetic field line encircling the Current Line;
the formula's denominator (2π d) is a circumference, which is the length of the field line thererence there.
. . . the numerator (μ I) is the total source current that pierces (goes thru) the disk Area within that circumference ... this is the current flux
Ampere's Law says that the weakening of the magnetic field is due to it being spread along this circumference s
=> Σ B·s (closed loop) = μ ΣI_piercing   ... .

Circular Current Loop concentrates its B-field inside the loop

add up Biot-Savart contributions (to making the magnetic field) from charges moving along a Loop
A Loop of current is all the same distance from the center , and all   μ₀ Q_sourcev   are 90° from the r direction
      (so sin(90°) =1) , the effective length is the current loop's actual circumference.
=> B_{Loop center} = μ₀ I / (2 R) .
      . . . a wire coil is usually treated as N loops , all in the same place ... use their average   1/R :
=> B_coil = N B_loop

no matter how strong it is in the center, which depends on the radius R of the physical loop ,
      the field must become μ₀ I / (2π r) where the distance from souce wire to field point becomes very small .
. . . notice that these expressions are the same at r = R/π , so most of the Area is fairly uniform field .

same flux up thru the inside of the loop (if current is counter-clockwise)
      as there is down around the outside of the loop - spread out over a large Area on the page, so B is weak
. . . magnetic field due to a current loop is the same as that due to a permanent "button" disk magnet
=> North pole on counter-clockwise side, South pole on clockwise side

regular spacing of loops makes Magnetic Field nearly uniform within a Current Solenoid (Helix) - - - - - - - - - -

treat a solenoid as a series of loops with uniform spacing - - - - - - - - - -
each loop carries current I (it's really the same wire in a helix)
. . . with n loops per meter, an "Amperian rectangle" from solenoid axis outward shows uniform B-field
      until it extends past the wire loop radius and is pierced by the current.
. . . then, each meter of solenoid length is pierced by n current loops ; Ampere says   B·1[m] = μ₀ n I . . .
=> B_{solenoid inside} = μ₀ n I . . . outside, the field is nearly zero.

What Else Does B DO? External Magnetic Field Applies Non-Zero Torque to a Current Loop - - - - - - - - - - - - -

Recall (from Physics 1) that Torque around an axis depends on the distance from the axis that the Force is applied ;
. . . τ = r × F   since the magnetic Force is perp. to B , large Torque occurs where B || r ;
      for a "rectangular" current loop, Force applied to length "L" ( |  to B) causes Torque due to width "W" ( || to B) .
=> τ = I A×B   ... where the loop Area A = L × W (or any other shape)

Recall (from Physics 1) that Torque can do Work as the loop orientation changes : W = τ· Δθ
      W = F·Δs , but in rotation , s is the arc length r Δθ along the circumference
. . . a current loop tries to align its interior B-field with the external magnetic field.
      in that sense, the current loop has a PE which varies with orientation
=> PE = − I A·B   . . . it is minimum when Area (thumb, when fingers curl along I) is || B .

Atoms tend to align their own magnetic field with an external field.
the valence electron "cloud" for many atoms can be treated as a current loop ; if it can re-orient,
. . . the successfully aligned magnetic fields add up . . . especially noticeable in a "ferromagnet" material
=> magnetic field inside material is usually more intense than outside . . . in iron, is 500× -to- 2000× as intense

The Lorentz Force ... qv×B ... Separates + from − charges . - - - - - - - - - -

Mobile Charges Separate - via qv×B - - - - - - - -
. . . until the charge accumulated at the conductor's edges   Q   causes an Electric Field
      whose Electric Force on any new charge q is strong enough to cancel the Magnetic Field Force.
. . . qE = −qv×B , so the Forces on the new charge cancels (no more accumulates at the edges) ... but
      E ≈ − v×B all through the conductor , so large voltages might develop. If a conductor with width δx has v along +y,
. . . δV = −E·δx = v×B ·δx   called "motional EMF" , .
      Notice that δx × v defines a   rate of Area swept by the motion .
. . . this can be used to generate DC voltage and current , by spinning a disk which is pierced by a magnetic field
If (instead) current is pushed through such a device , it forces the disk to spin . . . a DC motor .

The E-field caused by the Charges (distributed at the object's edges) is accompanied by a Voltage
=> δV = −E·δs   ... called the "Hall Voltage", when E is caused by current-carrying charges , separated by an external B .

Since only relative motion is meaningful, charge separation also occurs if a "moving" magnet is swept past a "stationary" conductor

Idea # 10 : the Magnetic Field imparts Energy to the Volume of Space it Occupies

It is this aspect that determines how the magnetic field lines spread out from some magnet :
. . . if they curl too tightly (from North pole to South pole), they would be very close together
      so B² would make a very large Energy density (even though in a small Volume).
. . . if they extend too far from the magnet, they would occupy a very large Volume
      even though the energy density would be very sparse .
=> Fields behave in a way that makes their Total Field Energy minimum . . . like any other Potential Energy .

Electric Fields and Gravitational Fields behave the same way ... see Physics 2's Topic 2 Summary.