Vector Analysis & Electromagnetics Flashcards
Three types of Vectors
Free, Sliding, Position
another term for the head endpoint of a vector
Terminus
How to evaluate the vector between two points
Vector = A - B
A - point (ax , ay , az)
B - point (bx , by , bz)
Vector AB = < (ax-bx) , (ay-by) , (az-bz)>
Formula for the Unit Vector(û)
û = u / (|u|)
u - Vector
|u| - Magnitude of the vector
Calcu: Applicable in VECTOR mode
- ) input vector into VctA
- ) input to calcu: VctA / |VctA|
Formula for the Magnitude of the vector(|u|)
|u| = sqrt(ux^2 + uy^2 +uz^2)
ux, uy, uz - vector components of vector u
Calcu: Applicable in VECTOR mode
- ) input vector into VctA
- ) input to calcu: |VctA|
Formula for the Midpoint of a Vector
Midpoint = Vector / 2
:v
CALCUTECH: Angle between 2 vectors
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)
The dot product of two vectors is a (Scalar/Vector)
Scalar
Formula for dot product
| A |, | B | - magnitude of 2 vectors
θ - Angle between two vectors
A | | B | cos θ
When vectors A and B are perpendicular to each other, what is the dot product of A and B?
0
The Cross Product of two vectors is a (Scalar/Vector)
Vector
Formula for the MAGNITUDE of the Cross product
| A |, | B | - magnitude of 2 vectors
θ - Angle between two vectors
A | | B | sin θ
(ANSWER HERE IS SCALAR MAGNITUDE)
How do you obtain the cross product? (Vector Answer)
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)
Is Cross Product product commutative?
no:
A X B) = -(B X A
Formula for area of parallelepiped formed by two vectors
A(paralellepiped) = |A X B|
Formula for area of triangle formed by two vectors
A(triangle) = A(paralellepiped) / 2 = |A X B| / 2
The third vector formed by the cross product of the two vectors is related to the two vectors in what way?
third vector forms a right angle to either of the two vectors (orthogonal to both)
When vectors A and B are parallel to each other, what is the cross product of A and B?
0
How to use Corkscrew method to determine direction of 3rd vector
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”
Formula for Scalar Projection
S = |A(dot)B| / |B|
or
S = A (dot) (Unit vector of B)
is scalar/vector projection commutative?
is scalar proj. of A to B equal to scalar proj. of B to A?
no
Formula for Vector Projection
V = |A(dot)B| B / (|B|)^2
or
V = (Scalar Projection) ( B / |B| )
The Result of Scalar Projection is a (Scalar/Vector)
Scalar
The Result of Vector Projection is a (Scalar/Vector)
Vector
Formula of Volume of a 3D Paralellepiped formed by 3 Vectors
V = |A (dot) (B X C)|
CALTECH: Formula of Volume of a 3D Paralellepiped formed by 3 Vectors
Use MATRIX Mode: V = Determinant of: | Ax Ay Az | | Bx By Bz | | Cx Cy Cz |
Formula for Volume of Tetrahedron
V = V(parallelepiped) / 6
If all three vectors are coplanar, What is the volulme of the parallelepiped formed?
0
What is a line integral used for?
To get the length of a curve in 3D Space
Formula for Line Integral
dL = dx i + dy j + dz k
note: reduce all variables into one variable to solve
Formula for Arc Length in 3D
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
Formula for Volume Integral using rectangular coordinate system
dV = dxdydz V = ∫∫∫dxdydz
Formula for Volume Integral using cylindrical coordinate system
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
Formula for Volume Integral using Spherical coordinate system
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
Formula for Line Integral using rectangular coordinate system
dL = dx i + dy j + dz k L = ∫dx i + ∫dy j + ∫dz k
Formula for Line Integral using cylindrical coordinate system
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
Formula for Line Integral using Spherical coordinate system
dL = (dr)i + (rsinθ . dφ)j + (rdθ)k L = ∫(dr)i + ∫(rsinθ . dφ)j + ∫(rdθ)k
What is the Point form for the Gradient Operator
∇U
Gradient operator Converts a (Scalar/Vector) into a (Scalar/Vector)
Scalar»»»»Vector
The Gradient is also used to obtain the ______ Vector
Normal
Formula for the Gradient (∇U)
∇U = ∂U/∂x + ∂U/∂y + ∂U/∂z
U is a scalar field in terms of x, y and z
Temperature is a (Scalar/Vector) Field
Scalar
What is the Point Form of the Divergence Operator
∇⋅a
Divergence operator Converts a (Scalar/Vector) into a (Scalar/Vector)
Vector»»»»>Scalar
What does the divergence of a vector measure?
It measures the flux/vectors entering an enclosed surface/volume, minus the flux/vectors going out of the surface/volume
Formula for the Divergence Operator
∇⋅a = ∂ax/∂x + ∂ay/∂y +∂az/∂z
a is a vector
ax, ay, az are the x y and z components of vector a
What is the Divergence of a Magnetic field H?
∇⋅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)
What is the Divergence of an Electric Field E?
∇⋅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)
What is the Point Form of a Laplacian Operator?
∇²U
Laplacian operator Converts a (Scalar/Vector) into a (Scalar/Vector)
Scalar»»»>Scalar
Formula for Laplacian Operator
∇²U = ∇⋅(∇U) = Div(Grad U)
What is the Point Form For the Curl Operator?
∇Xa
Curl operator Converts a (Scalar/Vector) into a (Scalar/Vector)
Vector»»»Vector
The Curl of a Vector measures __________
How much a field curls around a point
Formula for Curl of a Vector
∇Xa = | i j k | | ∂/∂x ∂/∂y ∂/∂z | | ax ay az | Perform Basket/Cofactor method answer is a vector
The Curl of an electric field (∇XE)
∇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)
The Curl of a magnetic Field(∇XH)
∇XH = J
J is the current density (Amperes / m^2)
When The Divergence of a Field (∇⋅F) Is equal to zero, the Field is said to be
Solenoidal/Divergenceless
When The Curl of a Field (∇XF) Is equal to zero, the Field is said to be
Irrotational/Conservative/Potential
Evaluate:
div(curl a)
div(curl a) = 0
Evaluate:
curl(grad U)
curl(grad U) = 0
Evaluate:
Curl(div a)
Invalid:
Div a will result into a scalar, and there is no such thing as a curl for a scalar
Evaluate:
grad(Curl a)
Invalid:
Curl a will result into a Vector, and there is no such thing as a gradient for a vector
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)
Stoke’s Theorem
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
Stoke’s Theorem
Also known as Gauss-Ostogradsky Theorem
Divergence Theorem
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)
Divergence Theorem
How to Represent Charges in Space using vectors
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
Formula for Vector from Q1 to Q2 (R12 and its unit vector, a12)
R12 = R2 - R1 a12 = R12 / | R12 | = (R2 - R1) / |R2 - R1|
Formula for Force exerted by one charge to another (F)
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
Formula for Electric Field exerted by a charge on a point in space (E)
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
Formula for The Charge of an enclosed volume (Q)
With Uniform Charge Density
Q = ρv ⋅ V
ρv - Volumetric Charge density (Coloumb / m^3)
V - Volume of charged surface
Formula for The Charge of an enclosed volume (Q)
Non-Uniform Charge Density
Q = ∫ρv ⋅ dV
ρv - Volumetric Charge density (Coloumb / m^3)
V - Volume of charged surface
Formula for E-Field emitted by a charge inside an insulating sphere
E = (k Q r) / R^3
r - point of interest
R - Sphere Radius
k - 9 x 10^9
Q - Charge in the sphere
Formula for the Electric field emitted by an Infinite Line of Charge
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
Formula for the Electric field emitted by an Infinite Sheet of Charge
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
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)
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
A proton moves ________ the E-Field Vector
along with
Anelectron moves ________ the E-Field Vector
against
E-Fields originate at the _____ charge
positive
E-Fields Terminate at the _____ charge
negative
The number of electric flux lines (ψ) is equal to ______
The number of charges (Q)
Formula for Electric Flux Density (D)
D = εE
ε - Permittivity (εrεo)
E - Electric Field Intensity
Unit of Electric Flux Density
D = Coloumb/m^2
The property in magnetism analogous to Electric Flux Density
Magnetic Flux density :v
B = μH
μ - Permeability
H - Magnetic Field Intensity
Formula for Flux density in the presence of a uniform Line Charge Density
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
Formula for Flux density in the presence of a uniform Sheet Charge Density
Dsheet = [ρs / 2 ] an
ρs - Areal Charge Density (Coloumb/m^2)
an -Unit Vector normal to the sheet charge
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
Gauss’ Law
A restatement of Gauss’ Law using Divergence Theorem
∇⋅E = ρv / εo
ρv - Volumetric Charge Density (Coloumb/m^3)
εo - Permittivity of Free Space (8.854 x 10^12)
Formula for Work of a Charge in the presence of another charge
W = KQ1Q2 / r W = QV
Define Voltage:
The work exerted per unit charge of the charge in motion
Formula for Voltage
V = kQ/r
Formula of Work In terms of Energy
W = ΔKE = -ΔPE W = 0.5mv^2 = -mgh
Four Helpful Equations in Elemag
W = QV V = Ed F = QE W = Fd
Theory: When a Current in a conductor is present, A magnetic Field is also present
Biot Savart Law
Formula for the magnetic field (H) surrounding an infinitely long straight current element
H = I / (2πr)
r - point of interest, distance perpendicular to current element
I - Current
Theory: If there is a magnetic Field loop
present, A current inside the loop must be present
Ampere’s Circuit Law
Right Hand Rule for Charged Particle’s Velocity in the presence of a magnetic field
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
Formula for Force on moving Charge due to E-Field
Fvector = Q (Evector)
Note: Mode VECTOR applicable
Formula for Force on moving Charge due to H-Field
F = QVB (sinθ) Fvector = Q(Vvector)x(Bvector)
V-Velocity
B-Magnetic Flux Density
Note: Mode VECTOR applicable
Formula for Force on moving Charge due to both E-Field and H-Field (AKA Lorentz Force)
Fvector = Q [ Evector + (Vvector)x(Bvector) ]
V-Velocity
B-Magnetic Flux Density
Note: Mode VECTOR applicable
Formula for Force on Current Carrying Conductor
F = BILsinθ =QVBsinθ F = I (Lvector X Bvector)
V-Velocity
B-Magnetic Flux Density
I - Current
L - Cable Length
Note: Mode VECTOR applicable
Force Between two Parallel wires carrying current
F = 2x10^-7 (I1 I2 L / d)
I1 and I2 - Currents in the wire
L - Length of Wire
d - Distance between wires
The Maxwell Law that also defines Gauss’ Law
∇⋅D = ρv
D - Electric Flux Density
ρv - Volumetric Charge Density (Coloumb/m^3)
The Maxwell Law that proves the Conservative nature of an electric field
∇XE = 0
The Maxwell Law that Defines Ampere’s Law
∇XE = J
J - Current Density (Ampere/m^2)
The Maxwell Law that Proves the non-existence of Magnetic Monopoles
∇⋅B = 0
