# Questions tagged [conservative-field]

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### How to prove that work due to conservative forces are independent of path?

The Wikipedia article on conservative forces says, A force field F, defined everywhere in space (or within a simply-connected volume of space), is called a conservative force or conservative vector ...
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### What is the vector field associated with potential energy?

The mere concept of a line integral is defined for a vector field, and I thus thought the following was a rigorous and general definition of potential energy: Definition: Given a conservative force ...
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### How do we visualise multiplication and division (reciprocal multiplication) in physical equations?

All formulas have terms generally multiplied or divided to represent another physical quantity. Like $F=ma$, $I=Q/t$, $W=F.s$ etc. Technically this is so because of ratios and proportionalities, the ...
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### Is David Tong incorrect in this remark about classical mechanics in his QM lectures?

In page 11 of his Quantum Mechanics lectures, we have the following quote: It turns out that not all classical theories can be written using a Hamiltonian. Roughly speaking, only those theories that ...
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### How to show that a radially symmetric central force is conservative?

Let $U\subseteq \mathbb{R}^3$ be open and $f:U\to\mathbb{R}^3$ be a radially symmetric central force, that is, a force field such that $$f(p) = -g(r)u_r$$ where $r=|p|$ and $u_r$ is the unit vector ...
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### Let $U$ be the potential energy associated with the force $F$. Why is $\frac{d}{dx}U=-F$?

In a conservative force field, we may define a function $U:\mathbb{R}^3\to\mathbb{R}$ such that $$\int_CFdx = U(x_A)-U(x_B)$$ and we call $U$ the potential energy associated with the force $F$. I've ...
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### Why is the force being the differential of a potential equivalent to it being a conservative force?

I was reading Goldstein's book on mechanics and came across this theorem: $F(r) = - \nabla V(r)$ is a necessary and sufficient condition of the force field being conservative. So far, I have ...
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### Does the Newtonian gravitational field have momentum analogous to the Poynting vector?

We can define the total energy of the electromagnetic field as: $$\mathcal{E}_{EM}= \frac{1}{2} \int_V \left(\varepsilon_0\boldsymbol{E}^2+\frac{\boldsymbol{B}^2}{\mu_0}\right)dV$$ which satisfies the ...
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### How to identify electrostatic field function out of some given functions? [closed]

Suppose I have a vector function $$\vec{v} = p(x,y,z) \ \hat{x} + q(x,y,z) \ \hat{y} + r(x,y,z) \ \hat{z}$$ How can I determine whether the given function represents an electrostatic field or not?
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### Fluid-mechanics: Scalar field associated with velocity field

I have just started studying fluid mechanics (without a proper physics education :) and came across the following equation for incompressible steady-state fluids. $$\nabla\cdot \mathbf{u} = 0$$ ...
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### Do optical tweezers provide conservative forces?

So, either it seems I am mislead with the idea of the conservative and non-conservative forces or I never knew/understood it. What exactly does the work in optical tweezers to make object levitating? ...
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### Why the change in gravitational potential energy of the two particle system remains same even when the both the masses are moving?

When we calculate Gravitational Potential energy of the two masses, we fix one mass and calculate the force acting on the other mass. Work done by the the force which is acting on the fixed mass is ...
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### Conservation and potential with non-cartesian forces

I understand how to determine if a force is conservative from \begin{equation} \nabla\times \mathbf{F}=0 \implies \mathbf{F}\text{ is conservative} \end{equation} When $F$ is in cartesian coordinates. ...
1 vote
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### How can a time-dependent gravitational field be conservative?

Let's consider 2 point particle graviting the one around the other. Can that gravitational field be considered conservative? I can go from A to B and then, after a time $\Delta t$ come back to A with ...
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### Conservative electric field must be static?

My question means, by Maxwell equations: $$\nabla\times \vec{E}=0\stackrel{?}{\implies} \frac{\partial \vec{E}}{\partial t}=0$$ I think that is right, this is my explanation, Intuitive explanation: A ...
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### Find the curl if the vector field depends on a parameter

Given the following vector, \begin{align} F(x(t),y(t),z(t)) &= \begin{bmatrix} \omega_1^2 x_o\cos(\omega_1 t) \\ \omega_2y_0\sin(\omega_2 t)\\ 0\\ \...
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### What is the property of a counterpart of conservative vector field in Minkowski space?

As we know, a conservative vector field is defined by a vector field $\vec {v} :U\to \mathbb {R} ^{n}$, where $\vec {v} =\nabla \varphi$. It is also an irrotational ...
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### Is net force conservative?

From the work-energy theorem, $$\int_{C}^{}\vec{F}\cdot d\vec{r}= \frac{1}{2}mv^2_f -\frac{1}{2}mv^2_i$$ Is velocity the gradient of position, and if so, does that make this force a conservative ...
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### Do action-reaction conservative forces come in pairs?

If $\ \vec{F_{12}}\$ denotes the force on particle 1 by particle 2 and is conservative, is $\ \vec{F_{21}}\$conservative too?
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### Interaction forces always depend on positions only through the distance, therefore conservative?

Suppose that two point masses $A_1,A_2$ are in interaction with each other, resulting in forces $F_1$ (acted upon $A_1$) and $F_2$ (acted upon $A_2$). Let $\bf{x}_1$,$\bf{x}_2$ be their respective ...
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### Relation between potential energy and conservative force

Does potential energy only happen when the work done is by a conservative force? Or does work done by non-conservative forces also create potential energy?
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### Correlation between conservative forces, non-conservative forces and potential energy

So I recently learned the definition of conservative forces, and how the work done by such forces depends only on the initial and final position of the particle but then we learnt about definition of ...
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### Do Electrostatic Charges build up at the ends of an Inductor in closed circuit?

I was watching this video from YaleCourses youtube channel. At around 41.00 minutes, the professor introduces the notion of charge buildup at the ends of an Inductor in a closed circuit. Is the ...
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