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Jahrdenas_GEAS > Vector Analysis & Electromagnetics > Flashcards

Vector Analysis & Electromagnetics Flashcards

1
Q

Three types of Vectors

A

Free, Sliding, Position

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2
Q

another term for the head endpoint of a vector

A

Terminus

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3
Q

How to evaluate the vector between two points

A

Vector = A - B

A - point (ax , ay , az)
B - point (bx , by , bz)

Vector AB = < (ax-bx) , (ay-by) , (az-bz)>

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4
Q

Formula for the Unit Vector(û)

A

û = u / (|u|)

u - Vector
|u| - Magnitude of the vector

Calcu: Applicable in VECTOR mode

  1. ) input vector into VctA
  2. ) input to calcu: VctA / |VctA|
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5
Q

Formula for the Magnitude of the vector(|u|)

A

|u| = sqrt(ux^2 + uy^2 +uz^2)

ux, uy, uz - vector components of vector u

Calcu: Applicable in VECTOR mode

  1. ) input vector into VctA
  2. ) input to calcu: |VctA|
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6
Q

Formula for the Midpoint of a Vector

A

Midpoint = Vector / 2

:v

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7
Q

CALCUTECH: Angle between 2 vectors

A 
In VECTOR mode:
1.) store the 2 vectors into VctA and VctB
2.) type equation and evaluate:
(VctA (dot) Vct B) / abs(VctA X VctB) 
3.) ArcCos(ANS)
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8
Q

The dot product of two vectors is a (Scalar/Vector)

A

Scalar

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9
Q

Formula for dot product

A

| A |, | B | - magnitude of 2 vectors
θ - Angle between two vectors

A | | B | cos θ

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10
Q

When vectors A and B are perpendicular to each other, what is the dot product of A and B?

A

0

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11
Q

The Cross Product of two vectors is a (Scalar/Vector)

A

Vector

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12
Q

Formula for the MAGNITUDE of the Cross product

A

| A |, | B | - magnitude of 2 vectors
θ - Angle between two vectors

A | | B | sin θ
(ANSWER HERE IS SCALAR MAGNITUDE)

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13
Q

How do you obtain the cross product? (Vector Answer)

A 
Use Basket Method on the matrix:
|Row 1 (i , j, k)|
|Row 2(Ax, Ay, Az)|
|Row 3(Bx, By, Bz)|
Or Cofactor Method (Pivot @ i, j, AND k)
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14
Q

Is Cross Product product commutative?

A

no:

A X B) = -(B X A

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15
Q

Formula for area of parallelepiped formed by two vectors

A

A(paralellepiped) = |A X B|

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16
Q

Formula for area of triangle formed by two vectors

A

A(triangle) = A(paralellepiped) / 2 = |A X B| / 2

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17
Q

The third vector formed by the cross product of the two vectors is related to the two vectors in what way?

A

third vector forms a right angle to either of the two vectors (orthogonal to both)

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18
Q

When vectors A and B are parallel to each other, what is the cross product of A and B?

A

0

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19
Q

How to use Corkscrew method to determine direction of 3rd vector

A

use right hand, let A be the fingers, B the palm, third vector is the thumb.
if AXB, A(fingers) approaches B(palm),
and the direction of the 3rd vector is how a screw(thumb) will move into the wall or out of the wall by using “Righty Tighty, Lefty Loosey”

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20
Q

Formula for Scalar Projection

A

S = |A(dot)B| / |B|

or

S = A (dot) (Unit vector of B)

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21
Q

is scalar/vector projection commutative?

is scalar proj. of A to B equal to scalar proj. of B to A?

A

no

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22
Q

Formula for Vector Projection

A

V = |A(dot)B| B / (|B|)^2
or
V = (Scalar Projection) ( B / |B| )

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23
Q

The Result of Scalar Projection is a (Scalar/Vector)

A

Scalar

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24
Q

The Result of Vector Projection is a (Scalar/Vector)

A

Vector

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25
Q

Formula of Volume of a 3D Paralellepiped formed by 3 Vectors

A

V = |A (dot) (B X C)|

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26
Q

CALTECH: Formula of Volume of a 3D Paralellepiped formed by 3 Vectors

A 
Use MATRIX Mode:
V = 
Determinant of:
| Ax  Ay  Az |
| Bx  By  Bz |
| Cx  Cy  Cz |
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27
Q

Formula for Volume of Tetrahedron

A

V = V(parallelepiped) / 6

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28
Q

If all three vectors are coplanar, What is the volulme of the parallelepiped formed?

A

0

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29
Q

What is a line integral used for?

A

To get the length of a curve in 3D Space

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30
Q

Formula for Line Integral

A

dL = dx i + dy j + dz k

note: reduce all variables into one variable to solve

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31
Q

Formula for Arc Length in 3D

A

given a curve in 3D:
a(t) = x(t) i +y(t) j + z(t) k ; x(t), y(t), z(t) are functions in terms of t

Length = ∫ sqrt( dx^2 + dy^2 + dz^2) dt

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32
Q

Formula for Volume Integral using rectangular coordinate system

A 
dV = dxdydz
V = ∫∫∫dxdydz
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33
Q

Formula for Volume Integral using cylindrical coordinate system

A 
dV = (ρdφ) dρ dz
V = ∫∫∫(ρdφ) dρ dz

ρ - Radius of cylinder
φ - angle formed by radius in xy plane (usually with respect to x axis)
z - level of the radius

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34
Q

Formula for Volume Integral using Spherical coordinate system

A 
dV = (rsinθ . dφ) (rdθ) dr
V = ∫∫∫(rsinθ . dφ) (rdθ) dr

r - Radius of Sphere
φ - angle formed by rsinθ in xy plane (usually with respect to x axis)
θ - Angle Formed by the radius with respect to the +z axis

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35
Q

Formula for Line Integral using rectangular coordinate system

A 
dL = dx i + dy j + dz k
L = ∫dx i + ∫dy j + ∫dz k
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36
Q

Formula for Line Integral using cylindrical coordinate system

A 
dL =  (dρ)i + (ρdφ)j + (dz)k
L = ∫(dρ)i + ∫(ρdφ)j + ∫(dz)k

ρ - Radius of cylinder
φ - angle formed by radius in xy plane (usually with respect to x axis)
z - level of the radius

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37
Q

Formula for Line Integral using Spherical coordinate system

A 
dL = (dr)i + (rsinθ . dφ)j + (rdθ)k
L = ∫(dr)i + ∫(rsinθ . dφ)j + ∫(rdθ)k
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38
Q

What is the Point form for the Gradient Operator

A

∇U

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39
Q

Gradient operator Converts a (Scalar/Vector) into a (Scalar/Vector)

A

Scalar»»»»Vector

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40
Q

The Gradient is also used to obtain the ______ Vector

A

Normal

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41
Q

Formula for the Gradient (∇U)

A

∇U = ∂U/∂x + ∂U/∂y + ∂U/∂z

U is a scalar field in terms of x, y and z

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42
Q

Temperature is a (Scalar/Vector) Field

A

Scalar

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43
Q

What is the Point Form of the Divergence Operator

A

∇⋅a

44
Q

Divergence operator Converts a (Scalar/Vector) into a (Scalar/Vector)

A

Vector»»»»>Scalar

45
Q

What does the divergence of a vector measure?

A

It measures the flux/vectors entering an enclosed surface/volume, minus the flux/vectors going out of the surface/volume

46
Q

Formula for the Divergence Operator

A

∇⋅a = ∂ax/∂x + ∂ay/∂y +∂az/∂z

a is a vector
ax, ay, az are the x y and z components of vector a

47
Q

What is the Divergence of a Magnetic field H?

A

∇⋅H = 0

(Imagine a magnetic field, and the bar magnet is your surface/volume. the number of flux lines leaving the north pole is equal to the number of flux lines entering the south pole, therefore, (Flux entering - Flux Leaving) = 0)

48
Q

What is the Divergence of an Electric Field E?

A

∇⋅H = ρv / εo

(An electric field emitted by a charge will always emit outward or always absorb inward, but not both, so no flux will try to enter back into the positive charge or be emitted by a negative charge, hence a non-zero Divergence)

49
Q

What is the Point Form of a Laplacian Operator?

A

∇²U

50
Q

Laplacian operator Converts a (Scalar/Vector) into a (Scalar/Vector)

A

Scalar»»»>Scalar

51
Q

Formula for Laplacian Operator

A

∇²U = ∇⋅(∇U) = Div(Grad U)

52
Q

What is the Point Form For the Curl Operator?

A

∇Xa

53
Q

Curl operator Converts a (Scalar/Vector) into a (Scalar/Vector)

A

Vector»»»Vector

54
Q

The Curl of a Vector measures __________

A

How much a field curls around a point

55
Q

Formula for Curl of a Vector

A 
∇Xa = 
|     i       j        k   |
| ∂/∂x  ∂/∂y  ∂/∂z |
|   ax    ay     az   |
Perform Basket/Cofactor method
answer is a vector
56
Q

The Curl of an electric field (∇XE)

A

∇XE = 0

(When a charge moves in an electric field, and returns to the initial point, regardless of the path taken by the point, the summation of the work exerted by/exerted to the particle is equal to 0)

57
Q

The Curl of a magnetic Field(∇XH)

A

∇XH = J

J is the current density (Amperes / m^2)

58
Q

When The Divergence of a Field (∇⋅F) Is equal to zero, the Field is said to be

A

Solenoidal/Divergenceless

59
Q

When The Curl of a Field (∇XF) Is equal to zero, the Field is said to be

A

Irrotational/Conservative/Potential

60
Q

Evaluate:

div(curl a)

A

div(curl a) = 0

61
Q

Evaluate:

curl(grad U)

A

curl(grad U) = 0

62
Q

Evaluate:

Curl(div a)

A

Invalid:

Div a will result into a scalar, and there is no such thing as a curl for a scalar

63
Q

Evaluate:

grad(Curl a)

A

Invalid:

Curl a will result into a Vector, and there is no such thing as a gradient for a vector

64
Q

The theory that states that the Surface integral of(the curl of the open surface) is equal to its Line integral
(Surface integral + Curl = Line Integral)

A

Stoke’s Theorem

65
Q

Theory: The Circulation of a vector around a closed path L is equal to the surface integral of (the curl of the vector over the open surface S bounded by L

A

Stoke’s Theorem

66
Q

Also known as Gauss-Ostogradsky Theorem

A

Divergence Theorem

67
Q

Theory: Flux outgoing from a closed surface is equal to the Volume Integral of the Divergence of the Flux
(Volume Integral + Divergence = Surface Integral Integral)

A

Divergence Theorem

68
Q

How to Represent Charges in Space using vectors

A

Q1 will be represented by R1, w/c is a POSITION VECTOR, only indicates position with respect to origin

Q2 will be represented by R2, w/c is a POSITION VECTOR, only indicates position with respect to origin

69
Q

Formula for Vector from Q1 to Q2 (R12 and its unit vector, a12)

A 
R12 = R2 - R1
a12  =  R12 / | R12 |  = (R2 - R1) / |R2 - R1|
70
Q

Formula for Force exerted by one charge to another (F)

A

F =[ (k Q1 Q2) / R^2 ] a12

k=9 x 10^9
R - distance between Q2 and Q1
a12 - direction/unit vector from Q1 to Q2

71
Q

Formula for Electric Field exerted by a charge on a point in space (E)

A

E =[ (k Qtest) / R^2 ] a12

Qtest - Charge that emits the electric field
k = 9 x 10^9
R - distance from the charge to the point of interest where electric Field intensity is to be taken
a12 - direction/unit vector from Qtest to point of interest R

72
Q

Formula for The Charge of an enclosed volume (Q)

With Uniform Charge Density

A

Q = ρv ⋅ V

ρv - Volumetric Charge density (Coloumb / m^3)
V - Volume of charged surface

73
Q

Formula for The Charge of an enclosed volume (Q)

Non-Uniform Charge Density

A

Q = ∫ρv ⋅ dV

ρv - Volumetric Charge density (Coloumb / m^3)
V - Volume of charged surface

74
Q

Formula for E-Field emitted by a charge inside an insulating sphere

A

E = (k Q r) / R^3

r - point of interest
R - Sphere Radius
k - 9 x 10^9
Q - Charge in the sphere

75
Q

Formula for the Electric field emitted by an Infinite Line of Charge

A

E = [ρL / (2πεo ⋅ r) ] ap

ρL - Linear Charge Density (Coloumb/m)
εo - Permittivity of Free Space (8.854 x 10^12)
r = Perpendicular distance, from the line charge to the Point of interest, where electric field intensity is to be sampled
ap -Unit Vector Perpendicular to the line charge

76
Q

Formula for the Electric field emitted by an Infinite Sheet of Charge

A

E = [ρs / (2εo) ] an

ρs - Areal Charge Density (Coloumb/m^2)
εo - Permittivity of Free Space (8.854 x 10^12)
an -Unit Vector normal to the sheet charge

77
Q

For The electric field emitted by an infinite sheet of charge, why is there no ‘r’ variable? (Distance from sheet charge to point of interest)

A

Assuming the sheet charge is infinitely long and wide, the E-Field Intensity anywhere in space is equal, so no need for ‘r’ variable in the equation

78
Q

A proton moves ________ the E-Field Vector

A

along with

79
Q

Anelectron moves ________ the E-Field Vector

A

against

80
Q

E-Fields originate at the _____ charge

A

positive

81
Q

E-Fields Terminate at the _____ charge

A

negative

82
Q

The number of electric flux lines (ψ) is equal to ______

A

The number of charges (Q)

83
Q

Formula for Electric Flux Density (D)

A

D = εE

ε - Permittivity (εrεo)
E - Electric Field Intensity

84
Q

Unit of Electric Flux Density

A

D = Coloumb/m^2

85
Q

The property in magnetism analogous to Electric Flux Density

A

Magnetic Flux density :v

B = μH

μ - Permeability
H - Magnetic Field Intensity

86
Q

Formula for Flux density in the presence of a uniform Line Charge Density

A

Dline = [ρL / (2π ⋅ r) ] ap

ρL - Linear Charge Density (Coloumb/m)
r = Perpendicular distance, from the line charge to the Point of interest, where Flux Density is to be sampled
ap -Unit Vector Perpendicular to the line charge

87
Q

Formula for Flux density in the presence of a uniform Sheet Charge Density

A

Dsheet = [ρs / 2 ] an

ρs - Areal Charge Density (Coloumb/m^2)
an -Unit Vector normal to the sheet charge

88
Q

Theory: The Flux lines that leave an enclosed surface/volume is equal to the number of charges enclosed divided by the permittivity of free space

A

Gauss’ Law

89
Q

A restatement of Gauss’ Law using Divergence Theorem

A

∇⋅E = ρv / εo

ρv - Volumetric Charge Density (Coloumb/m^3)
εo - Permittivity of Free Space (8.854 x 10^12)

90
Q

Formula for Work of a Charge in the presence of another charge

A 
W = KQ1Q2 / r
W = QV
91
Q

Define Voltage:

A

The work exerted per unit charge of the charge in motion

92
Q

Formula for Voltage

A

V = kQ/r

93
Q

Formula of Work In terms of Energy

A 
W = ΔKE = -ΔPE
W =  0.5mv^2 = -mgh
94
Q

Four Helpful Equations in Elemag

A 
W = QV
V = Ed
F = QE
W = Fd
95
Q

Theory: When a Current in a conductor is present, A magnetic Field is also present

A

Biot Savart Law

96
Q

Formula for the magnetic field (H) surrounding an infinitely long straight current element

A

H = I / (2πr)

r - point of interest, distance perpendicular to current element
I - Current

97
Q

Theory: If there is a magnetic Field loop

present, A current inside the loop must be present

A

Ampere’s Circuit Law

98
Q

Right Hand Rule for Charged Particle’s Velocity in the presence of a magnetic field

A

Index Finger - If proton, Velocity points towards the index finger, if Electron, Velocity opposes direction of the index finger
Middle Finger - B Field/H Field
Thumb - Force

99
Q

Formula for Force on moving Charge due to E-Field

A

Fvector = Q (Evector)

Note: Mode VECTOR applicable

100
Q

Formula for Force on moving Charge due to H-Field

A 
F = QVB (sinθ)
Fvector = Q(Vvector)x(Bvector)

V-Velocity
B-Magnetic Flux Density

Note: Mode VECTOR applicable

101
Q

Formula for Force on moving Charge due to both E-Field and H-Field (AKA Lorentz Force)

A

Fvector = Q [ Evector + (Vvector)x(Bvector) ]

V-Velocity
B-Magnetic Flux Density

Note: Mode VECTOR applicable

102
Q

Formula for Force on Current Carrying Conductor

A 
F = BILsinθ =QVBsinθ
F = I (Lvector X Bvector)

V-Velocity
B-Magnetic Flux Density
I - Current
L - Cable Length

Note: Mode VECTOR applicable

103
Q

Force Between two Parallel wires carrying current

A

F = 2x10^-7 (I1 I2 L / d)

I1 and I2 - Currents in the wire
L - Length of Wire
d - Distance between wires

104
Q

The Maxwell Law that also defines Gauss’ Law

A

∇⋅D = ρv

D - Electric Flux Density
ρv - Volumetric Charge Density (Coloumb/m^3)

105
Q

The Maxwell Law that proves the Conservative nature of an electric field

A

∇XE = 0

106
Q

The Maxwell Law that Defines Ampere’s Law

A

∇XE = J

J - Current Density (Ampere/m^2)

107
Q

The Maxwell Law that Proves the non-existence of Magnetic Monopoles

A

∇⋅B = 0

Jahrdenas_GEAS (9 decks)

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