Jan 19, 2016

Chapter 33 & 34 Review

Chapter 33: The Magnetic Field

Cyclotron MotionNewtons LawBoth a and F directed toward center of circleEquate

What is the force on those N electrons?Force on a single electronForce on N electronsNow useLets make l a vectorpoints parallel to the wire in the direction of the current when I is positiveForce on wire segment:

Comments:Force is perpendicular to both B and lForce is proportional to I, B, and length of line segment

Superposition: To find the total force on a wire you must break it into segments and sum up the contributions from each segment

Electric FieldMagnetic FieldqWhat are the magnitudes and directions of the electric and magnetic fields at this point? Assume q > 0 rComparisons: both go like r-2, are proportional to q, have 4p in the denominator, have funny Greek lettersDifferences: E along r, B perpendicular to r and v

Magnetic Field due to a currentMagnetic Field due to a single chargeIf many charges use superposition#2 q2 , v2#1 q1, v1#3 q3 , v3r3r2r1Where I want to know what B is

For moving charges in a wire, first sum over charges in each segment, then sum over segments

Summing over segments - integrating along curveIrIntegral expression looks simple but..you have to keep track of two position vectorsBiot Savart lawwhich is where you want to know Bwhich is the location of the line segment that is contributing to B. This is what you integrate over.

Magnetic field due to an infinitely long wirexzICurrent I flows along z axisI want to find B at the pointI will sum over segments at points

rcompare with E-field for a line charge

Gauss Law:Biot-Savart Law implies Gauss Law and Amperes LawBut also, Gauss law and Amperes Law imply the Biot -Savart lawAmperes Law

Electric field due to a single chargeMagnetic field due to a single loop of currentGuassian surfaces

# turns per unit length

Gauss Law:Biot-Savart Law implies Gauss Law and Amperes LawBut also, Gauss law and Amperes Law imply the Biot -Savart lawAmperes Law

Chapter 34: Faradays Law of Induction

Faradays Law for Moving Loops

Magnetic FluxSome surfaceRemember for a closed surfaceMagnetic flux measures how much magnetic field passes through a given surfaceOpen surfaceClosed surface

Suppose the rectangle is oriented do that are parallelRectangular surface in a constant magnetic field. Flux depends on orientation of surface relative to direction of B

Lenzs LawIn a loop through which there is a change in magnetic flux, and EMF is induced that tends to resist the change in fluxWhat is the direction of the magnetic field made by the current I?

Into the pageB. Out of the page

Reasons Flux Through a Loop Can ChangeLocation of loop can change

Shape of loop can change

Orientation of loop can change

Magnetic field can change

Faradays Law for Moving LoopsFaradays Law for Stationary Loops

LVBRVRVLIINow I have cleaned things up making use of IB=-IR, IR=IL=I.

Now use device laws:VR = RIVL = L dI/dtKVL: VL + VR - VB = 0This is a differential equation that determines I(t). Need an initial condition I(0)=0

Solution:Lets verifyThis is called the L over R time.

What is the voltage across the resistor and the inductor?

LVBRVRVLIIInitially I is small and VR is small.All of VB falls across the inductor, VL=VB.Inductor acts like an open circuit.Time asymptotically I stops changing and VL is small.All of VB falls across the resistor, VR=VB. I=VB/RInductor acts like an short circuit.

Lets take a special case of no current initially flowing through the inductorSolutionA:B:Initial charge on capacitor

Current through Inductor and Energy StoredEnergyt

Foolproof sign convention for two terminal devicesLabel current going in one terminal (your choice).Define voltage to be potential at that terminal wrt the other terminalV= V2 -V13. Then no minus signsPower to deviceKVL Loop

Contribution to voltage sum = +V

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