Lecture 32: Conservative Fields

What is the definition of “Work” and the equation?

sum of force vector “F” dot product dl(displacmenr vector), integral b to a

What is needed to use the potential energy method?

conservative force (gravity, Hooke’s law, electrostatic force)

Criteria 1 for conservative forces

Path Independence - the total work done going from point A

to point B is the same regardless of the path taken there

Criteria 2 for conservative forces

Potential - The force vector can be expressed in terms of derivative(s) of a potential, which we refer to as potential energy U

Criteria 3 for conservative forces

Irrotational - The “curl” of the force vector yields the zero vector

Conservative force in terms of potential energy

F = -lambert U

Potential energy as integral of force

-U = integral b to a F * dl

what is F*dl representing

dot product of direct of objects displacement w force acting on it

positive dot product?

vectors are at less than 90 (same direction) - work is +

negative dot product?

vectors are at more than 90 (opposite direction) - work is -

as angle between vectors inc, work..?

gets more negative

forces “helping”’ the motion do ______ work?

positive

If we have a conservative force, the work is equal to the ______

change in potential energy

negative

Formula for work

W = -U = F*dl

fundamental theorem for line integrals definition

line integral of a gradient can

be expressed as the evaluation at the bounds of that functio

fundamental theorem for line integrals formula

integral U dl = Ub - Ua ( = integral F * dl)

total work formula

W = W a to b + W b to a = -(Ub - Ua) - (Ua - Ub) = 0

Total work over a closed loop

0

Irrotational Formula Condition

lambert x F = (0,0,0)

Stokes Theorem

any closed path integral can be related to the curl added up of the surface between the path, converts integral form of amperes law and faradays law → differential curl equations

formula for work due to stokes theorem

W = F*dl = double integral (lambert x F) * dA = 0

divergence theorem formula

double integral F dA = triple integral (lambert * F0 dV

what does divergence theorem do

replace a flux integral over a closed (e.g., Gaussian) surface with an integral of the divergenc

Examples of conservative fields

Newtonian gravitational field g and electrostatic field E

conservative fields critera 1

Path Independence - the path integral from point A to point

B is the same regardless of the path taken ther

conservative fields critera 2

Potential - The field vector can be expressed in terms of deriva-

tive(s) of a potential, which we refer to as scalar potential V

(electrostatics) or scalar potential ϕ (Newtonian gravity)

conservative fields critera 2

Irrotational - The “curl” of the vector field = 0

scalar potential formula (gravity)

-o = integral g*dl

V (electrostatics) formula

-V = integral E*dl

field written as negative gradient of scalar potential (gravity)

g = -del (o)

field written as negative gradient of V (electrostatics)

E = -del V

curl of each field must be zero to be irrotational,

giving the conditions __

del x g = 0,0,0 and del x E = 0,0,0

Gauss’s law for the gravitational field formula

double integral g*dA = -4piGM

Gauss’s law for the gravitational field definition

relates the total flux of the gravitational field over a closed

Gaussian surface to the mass inside of that closed surface

gauss’s law converted to differential form using divergence theorem (gravitational field)

del*g =-4piGp

gaussian sphere, if point charges are positive, electric field vectors are pointing ___

outwards (same as dA vectors)

Newtonian gravity case , point charges are _____, field vectors point _____ dA vectors point _____

negative, inward, outward

electrostatics Gauss’s law, e expressed in differential form using a divergence: formula

del * E = p/eo

gravitational Gauss’s law, e expressed in differential form using a divergence: formula

del * g = -4piGp

electrostatics formula Gauss’s law, differential equations in terms of scalar potentials

del * (delV) = -p/eo = laplacianV = -p/eo

gravitational formula Gauss’s law, differential equations in terms of scalar potentials

del * (delo) = 4piGp = laplacian o = 4piGp

electrostatic Gauss’s laws in terms of potential - poissons equation form

laplacianV = -p/eo

gravitational Gauss’s laws in terms of potential - poissons equation form

laplacian o = 4piGp

in poissons equation form, when laplacian of potential energy = 0, we have _____

Laplace’s equation

why is the theory describing electrostatics and the theory describing Newtonian gravity are virtually identical!

nature of conservative vector fields

are magnetic fields conservative?

no, they are rotational

Magnetic fields are _____-

solenoidal

solenoidal

zero divergence( del*B = 0)

magnetic vector potential A such that:

B = del x A