Question

conservative fields and gradient

Answer 1

what is the meaning of gradient, potential, and force... physically

Answer 2

i get everything about line integrals and work but im finding it hard to relate these 3 concepts still

Answer 3

Force is the interaction of an object with its environment; which causes a change in the magnitude and/or direction of its velocity.

Field is one way of describing the effects of a force (the other is action at a distance)

Potential Energy is a form of ordered energy an object possess due to its location in a conservative field.

Potential quantifies the strength of a field using scalars.

A field is conservative if potential is a state function.

Gradient is the change of potential over space.

Answer 4

potential is a property of conservative fields is it ?

Answer 5

v = dr/dt

a = dv/dt

F = ?
Potential = ?

Answer 6

i mean still its foggy... like gradiant gives slope along each axis right ?

Answer 7

wat does the phrase 'gradiant of a potential' mean ?

Answer 8

"potential is a property of conservative fields is it ?"
Yes, we cannot define potential for a non-conservative field

v = dr/dt
a = dv/dt
F = <mass> * dv/dt
Potential = a <dot> dx

If the field is non-conservative; we won't get a single value of potential for a point.

"...like gradiant gives slope along each axis right ?"
Can you describe which gradient are you talking about ??

Answer 9

ahh that makes sense so when we speak potential, we're talking about conservative fields thanks that clears up somethings xD

Answer 10

i know one gradiant : <fx, fy, fz>

Answer 11

partials of a funciton with respect to x, y and z

Answer 12

problem with me is im very weak in physics...

Answer 13

is potential same as energy ?

Answer 14

if so the order is like this :-

energy
force
acceleration
velocity
displacement
position

Answer 15

taking integral of the previous funciton gives the next function..

Answer 16

humm idk im getting super confused

Answer 17

The gradient you are talking about is a potential gradient and the function is the potential of the field.

Partial differentiation of potential function gives the field strength vector (another way of quantifying the strength of field, this time using vectors)

"is potential same as energy?"
Potential energy is described as the interaction between the parts of a system. Hence, we cannot assign potential energy to any particular object.

The concept of potential is like polymorphism -
E.g. Gravitational potential ≡ Potential energy per unit mass in the field (generated by some object)
Similarly Electric potential ≡ Potential energy per unit charge in the filed (generated by some object)

Answer 18

ohk so there is no magnetic potential as magnetic fields are not conservative

Answer 19

Albeit gravitational force (and electric force) is the interaction between two objects (lets say A and B); but we use the concept of potential to study all the interactions of type -
A and B
A and C
A and D
etc.

Answer 20

got that part, when we define field we talk about one object only

Answer 21

combining all 3 terms together :

gradiant defines how quickly the potential is changing in the given Force field.
is that correct to think ?

Answer 22

"ohk so there is no magnetic potential as magnetic fields are not conservative"
Magnetic field is also a conservative field (actually, electric and magnetic field are collectively called electromagnetic field; it just depends on the frame of reference what you observe)

We dont use the term "magnetic potential" because magnetic field can never do work on an object (as it is always directed perpendicular to the velocity).

"got that part, when we define field we talk about one object only"
Yes, and we replace the other object with a test object (like test mass or test charge)

"gradiant defines how quickly the potential is changing in the given Force field.(with respect to space; NOT time)"

Answer 23

OMG ! yes with respect to space... !!!

Answer 24

how far/near the level curves are = gradient

Answer 25

makes more sense now thanks a lot guess ill take a bit mroe time to fully digest these...

Answer 26

one last q..
closed loop line integral for magnetic fields is not 0 right ?
how can we say they're conservative

Answer 27

"if so the order is like this :-
energy
force
acceleration
velocity
displacement
position
taking integral of the previous funciton gives the next function.."

If you can remember the formulas relating these or their respective units; you can simply use -
multiplication ⇒ integration
division ⇒ differentiation

For ex. Work = Force * displacement ⇒ W = ∫F.dx
Acceleration = velocity / time ⇒ a = dv/dt

Answer 28

"closed loop line integral for magnetic fields is not 0 right ?
how can we say they're conservative"

Can you write the integral here ??

Answer 29

got it completely

Answer 30

in simple terms,
if i walk from point A to point B and return back to point A, I'll be doing some non-zero work if i am in magnetic field right ?

Answer 31

where as in gravitatinal/electric field the work will be 0 if i come back to start position

Answer 32

F = <-y, x>
im not sure if we can call this magnetic field ?... but clearly this is not conservative .. .

Answer 33

"if i walk from point A to point B and return back to point A, I'll be doing some non-zero work if i am in magnetic field right ?"

Even if you are somehow carrying an excess charge, magnetic field will not do any work on you (or in other words, you cannot do any work against the magnetic field) irrespective of whether you return to the same point or not (closed loop is not necessary here).

Still I'll prefer to see what closed loop integral you're working with..

"F = <-y, x>"
Yes, this is a non-conservative field, as the work done depends on the path (or in other words, we need a relation between y and x to calculate work)

In general, we have a formula to check whether a field is conservative or not..let me see if I can find it..

Answer 34

Euler's law
If f(x,y) is a state function; then
\[\frac{ ∂ }{ ∂y }\left( \frac{ ∂f }{ ∂x } \right)=\frac{ ∂ }{ ∂x }\left( \frac{ ∂f }{ ∂y } \right)\]

Now, if potential is a state function, then the field is conservative. :)