Final Exam May 8

6 problems, multiple pieces. Covers everything on homework and assigned chapters.

Study Tips: •  Read the book and take notes! •  Do as many problems as possible and understand them!

•  Problems from the Homeworks •  Problems worked out in the book •  Problems in other textbooks

•  Try simple problems first, work towards harder ones •  Don’t waste time memorizing everything •  Unfortunately, physics takes a lot of time and work

Topics: •  Charge and Induction

•  Conductors and Insulators •  Coulomb’s Law

•  Discrete charges: force direction and magnitude •  Continuous charges: integrating charge densities

•  Electric Field •  Calculation of direction and mag. with Coulomb’s Law •  Field lines---direction and shape •  Gauss’s Law: calculation of E-field •  Gauss’s Law: How do flux and field differ?

•  Electric Potential and Potential Energy •  Calculation of energy to assemble system of charges •  Direction charges move in a potential •  Potential of a continuous charge distribution •  Calculation of potential differences from E-field integration

•  Capacitors •  Calculating capacitance from Q=CΔV and integral of E-field between plates/spheres/cylinders…. •  Effects of a dielectric on C •  Parallel and series combinations •  Energy storage

•  Resistor Circuits •  Resistance and resistivity •  Power and power dissipation •  Potential at various points •  Current through resistors •  Multi-loop circuits

Topics:

Topics: •  Magnetic Force, magnitude and direction

•  On a moving charge •  Between wires •  On a rectangular loop •  Crossed E and B fields, velocity selectors •  Circulating charge in a B field

•  Biot-Savart Law: •  Calculate a magnetic field

•  Ampere’s Law •  Various geometries: solenoids, sheets of current, toroids…. •  Can it be used in a particular situation or not? •  Ampere’s Law with displacement current

•  Torque on a loop

•  magnetic dipole moment •  Force on wires

•  Faraday’s Law •  Changing B, area, direction, to get EMF •  Rotating loops •  Lenz’s Law and direction of current from emf

•  Inductors •  Calculating inductance •  Back EMF

•  RC, LR, LC Circuits •  Time constants and graphs •  Switches and currents and potentials •  Design a circuit? •  Frequency of oscillation •  Induction of EMF or EM waves?

Topics:

•  Maxwell’s Equations

•  Displacement current and Ampere’s Law •  What does each equation tells about physics?

•  EM Waves •  Transmission of power •  Direction of E, B and wave •  Wavelength, frequency, period, speed •  Index of refraction •  Poynting vector •  Intensity

Topics:

12212

21

012 ˆ

41 r

rqqF

πε=

!

ii

ii r

rqqF 121

1

01 ˆ

41πε∑=

!

Force

!E ⋅d!a = qenc

ε0=ΦE!∫

Field

i

in

i r

qV ∑ ==

104

1πε

rdq

dV04

1πε

=

Potential

ififif WUUU −=−=Δ

Potential Energy

0qFE!

!=

EqF!!

0=

xVEx ∂

∂−= ΔVab =

!E ⋅d!s

i

f

∫

0qUV Δ

=Δ

VqU Δ=Δ 0

∫ ⋅−=Δf

iif sdFU !!

xUFx ∂

∂−=

d!E = 1

4πε0dqr2r̂

)( BvqFB!!!

×=

)( BLiF!!!

×=

Force

dtdisdB E

netϕ

εµµ 000 +=⋅∫!!

Field

Potential Potential Energy

Junior year/grad school

Junior year/grad school, except for potential energy of a dipole in a field

20 ˆr

rsidBd ×=

!! µ

12212

21

012 ˆ

41 r

rqqF

πε=

!

€

! F NetQ

=∑ 14πε 0

Qqi

riQ2 ˆ r iQ

Coulomb’s Law Summary For just two discrete, point charges:

Where is the force of charge 1 ON charge 2

€

! F 12

€

! r 12 is the vector FROM charge 1 TO charge 2

€

ˆ r 12 =! r 12

| ! r 12 |Is the unit vector (magnitude=1.0!) in direction of

€

! r 12

For several discrete, point charges, acting on charge Q:

Step 4: Plug in to component equations in Step 2, using

Coulomb’s Law Summary

Step 3: Calculate magnitude of r-vector and find angle(s). In above example:

€

r1Q =| ! r 1Q |= x02 + y0

2

€

r2Q =| ! r 2Q |= x0

Make sure to include signs of vectors AND charges correctly! And check units!

Step 1: Draw all forces acting on charge in question:

+Q

q1=-q

q2=+q x0

y0

€

! F 1Q

€

! F 2Qα

Step 2: Write down components of each force, and add (e.g., in example above):

€

FNetQx = F1Qx+ F2Qx

= −F1Q cosα + F2Q Watch out for signs!! If component is `down’ or `to the left,’ it is negative! Watch out for angle: make sure you have correct trig functions (depends on which axis angle is relative to). If there is an `obvious’ symmetry that cancels components, you can ignore those.

€

FNetQy = F1Qy+ F2Qy

= −F1Q sinα

€

cosα =x0

x02 + y0

2

€

sinα =y0

x02 + y0

2

FiQ =14πε0

| qi ||Q |riQ2

Notice distance (r) is always TOTAL distance, regardless of component

`Force component’ method

(You are responsible for ensuring signs are correct!)

Step 4: Plug in to then add components, calc. mag. and dir.

€

! F NetQ

=∑ 14πε 0

Qqi

riQ2 ˆ r iQ

Coulomb’s Law Summary

Step 1: Draw all r-vectors between charges exerting force and charge feeling force:

+Q

q1=-q

q2=+q x0

y0

€

! r 1Q

€

! r 2Q

Step 2: Write down each r-vector in components, e.g. in the example above we have

€

! r 1Q = x0ˆ i + y0

ˆ j

€

! r 2Q = x0ˆ i

Step 3: Calculate magnitude of r-vector, and get the unit vector. In above example:

Watch out for signs!! If component is `down’ or `to the left,’ it is negative! If there is an `obvious’ symmetry that cancels components, you can ignore those, but be careful about the next step.

€

r1Q =| ! r 1Q |= x02 + y0

2

€

r2Q =| ! r 2Q |= x0

€

ˆ r 1Q =! r 1Q

| ! r 1Q |=

x0

x02 + y0

2ˆ i + y0

x02 + y0

2ˆ j

€

ˆ r 2Q = +ˆ i

Make sure to include signs of vectors AND charges correctly!

`Unit-vector’ Method

€

! E 1 =

14πε 0

q1

r12 ˆ r 1

€

! E Net =∑ 1

4πε 0

qi

ri2 ˆ r i

Electric Field Summary For a single discrete charge:

Where is the electric field of charge 1

€

! E 1

€

! r 1 is the vector FROM charge 1 TO point where field is being evaluated

€

ˆ r 1 =! r 1

| ! r 1 |Is the unit vector (magnitude=1.0!) in direction of

€

! r 1

For several discrete, point charges:

To calculate the force on a charge Q from an electric field, we use

€

! F NetQ = Q

! E Net

Field lines come radially OUT from + charges and go radially IN to – charges.

Electric Field Summary

Step 1: Write down the field due to a small element of charge (dq), for example:

Step 2: Change element of charge dq into something you can integrate over:

€

! r = x ˆ i + yˆ j

Step 3: Integrate---in example above this would be for x-component:

€

ˆ r =

! r | ! r |

=x

x 2 + y 2ˆ i + y

x 2 + y 2ˆ j

Continuous Distributions

y

L

dq

€

d! E = 1

4πε 0

dqr2 ˆ r

x

y r

€

! E x (y) =

14πε 0

λxdx(x 2 + y 2)3 / 20

L∫

So:

(This is a lot like Steps 1+2 of Coulomb’s Law Problems) Check to see if there are any symmetries that eliminate a component!

€

dq = λdx

λ =QL

For a line of charge

€

dq =σdA

σ =QArea

€

dq = ρdV

ρ =Q

Volume

For a charged surface For a charged volume

€

d! E =

14πε 0

dqx 2 + y 2 ( x

x 2 + y 2ˆ i + y

x 2 + y 2ˆ j )

Check units on integrand! (And include dx in the units!)

Bear in mind what is constant (y above) and what is varying (x above)