17 votes
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Contradiction in my understanding of wavefunction in finite potential well

Even classically, particles with a fixed total energy spend more time near the turning points since this is where the motion is the slowest. The probably of finding the particle in a small region ...
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  • 38.7k
15 votes

What is the difference between electrostatic and electric potential?

They're usually considered identical when speaking, but there are some details to keep in mind. Electrostatic potential is defined because, with the static regime hypothesis: $$\vec{\nabla}\times\vec{...
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  • 2,236
12 votes

Contradiction in my understanding of wavefunction in finite potential well

Imagine a perfectly elastic ball dropped vertically onto a flat surface. The ball heads for the point of lowest potential, ie the ground, but because of conservation of energy it bounces back to its ...
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6 votes
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What does negative electrical energy signify?

Absolute values of the potential energies of systems do not have any physical meaning. It is the change in potential energy that has a physical meaning. When the potential energy of a dipole system is ...
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  • 2,880
6 votes

Relation Between Gravitational Potential and Acceleration due to gravity

$V \ne -g$. For one thing, the gravitational potential is a scalar while $\vec g$ is a vector field, so they can't be equal. They aren't equal in magnitude either: $$ V = -\frac{GM}{r}$$ $$ \vec g = -\...
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  • 7,654
6 votes
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Electric potential generated by spherical symmetric charge density

The formula $$\phi(r)= \frac{1}{r}\int_0^r 4\pi\rho(r')r'^2dr' \tag{1}$$ for the potential is indeed wrong, as you have already proven by checking $\mathbf{E}(\mathbf{r})=-\nabla\phi$. It misses the ...
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5 votes

What is the difference between electrostatic and electric potential?

The electric potential and electrostatic potential are different words for the same electric phenomena in a circuit. It is defined as the amount of work energy needed to move a unit of electric charge ...
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5 votes
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How to solve the Gauss' law?

You can have a non-zero potential -- as long as it is constant in space. This will always generate a zero electric field, since in this case $\mathbf{E} = -\nabla\phi = 0$, which is indeed expected ...
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  • 296
4 votes

"Forbidden nodes" of a quantum particle trapped in a harmonic oscillator potential

There is a fundamental misconception here. When the wavefunction is a solution to the time-independent Schrödinger equation, the wavefunction yields a time-independent probability density: there is no ...
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4 votes
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What is the point of the four different thermodynamic potentials?

An an example, imagine that you have an insulated cylinder with a locked piston which is partially filled with liquid H$_2$O. Some amount of the liquid will evaporate and form an H$_2$O vapor ...
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4 votes

What is the difference between electrostatic and electric potential?

Usually They both are same. There is no difference, while studying electrostatics sometimes mentors prefers to use the word Electrostatic potential instead of saying Electric potential else both are ...
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4 votes
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How to find expansion of slightly modified Coulomb's potential?

The usual multipole expansion follows from the Legendre identity $$ {\displaystyle {\frac {1}{\sqrt {1-2xt+t^{2}}}}=\sum _{n=0}^{\infty }P_{n}(x)t^{n}} $$ The generalization to arbitrary powers ...
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3 votes

Electric potential generated by spherical symmetric charge density

The first formula seems to assume that the potential at point r inside the sphere is given only by the charge included in the sphere of radius r. This is true for the electric field but not for the ...
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  • 6,568
3 votes

Electric potential generated by spherical symmetric charge density

The first formula is true only far away from the source. It is in fact the first term of the multipole expansion. By using Gauss theorem, you obtain the correct result for the electric field, but if ...
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3 votes

"Forbidden nodes" of a quantum particle trapped in a harmonic oscillator potential

You can’t use the uncertainty principle to evaluate an arbitrary subset of a wavefunction. The uncertainty principle applies to the solutions of the Schrödinger equation. Specifically, the ...
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  • 71k
3 votes

What is the point of the four different thermodynamic potentials?

It's mostly about the conditions that must be satisfied to reach equilibrium. Given a specific transformation, you build the appropriate potential that will be minimized when equilibrium is reached. ...
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  • 2,236
3 votes

"Forbidden nodes" of a quantum particle trapped in a harmonic oscillator potential

In order to make sure I understood you correctly let me rephrase your question: why is the probability density of a particle equal to zero (at nods) even though there is nothing physically preventing ...
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3 votes

Contradiction in my understanding of wavefunction in finite potential well

You are probably looking at an energy eigenfunction, a wavefunction which has a definite energy. The statement "$E-V$ is the kinetic energy" does not apply to a single position but to the ...
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3 votes
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Why only the first derivative of the wave function must be continuous?

You don't have to enforce the continuity of $\psi''$. The time-independent Schrödinger equation is $$\psi''(x) = - \frac{2m\left[E-V(x)\right]}{\hbar^2}\psi(x).$$ Since the potential $V$ is continuous,...
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  • 7,654
2 votes

Analytical solution to Poisson's equation for gravity

I am studying Poisson's equation for gravity. $$\nabla^2 \varphi = 4\pi G\rho$$ I have read that it is solved analytically using some Green's function The intuition behind Green's functions is that ...
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2 votes
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Vector potential due to a spinning spherical shell with a non-uniform surface charge distribution

You are missing the $\hat\phi \cdot \hat \phi'$ factor in the integrand. That is, the current is along $\hat \phi'$, and the resulting vector potential is along $\hat \phi$, but these are two ...
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  • 1,897
2 votes

Relation between Electric Potential and Electric Field Intensity

Well, you called $y$ the function $V(x)$ which makes my answer awkward. We live in a three dimensional world, with coordinates, say $x,z,w$ since I cannot use $y$... As you wrote $dV = \vec{E}\cdot \...
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  • 3,361
2 votes

How to solve the Gauss' law?

Ive been unfortunately trying to apply simple boundary conditions on your equation with no luck, because it's wrong. so First of all, let me correct you: Laplaces equation for $r-$dependance is: $$\...
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  • 4,243
1 vote
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Derivation of the Yukawa potential by integration

One can use the identity $\nabla^2(\frac{1}{r})=-4\pi\delta(\vec{r})$. In other words, substitute $\nabla^2\Phi=\frac{1}{4\pi\epsilon_0r}\nabla^2g(r)-\frac{g(r)}{\epsilon_0}\delta^{(3)}(\vec{r})$ in ...
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  • 715
1 vote

Finding capacitance of this setup

Your school teacher is incorrect. Modeling the bottom dielectric as a single capacitor as in your last diagram assumes that the plane midway between the capacitor plates is an equipotential surface: i....
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  • 7,654
1 vote

Metal/good conductor frame in a changing magnetic field

according to Maxwell's equations, the changing magnetic field can produce non-conservative electric field which make electrons move along with the conductor. this non-conservative electric field is ...
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1 vote

Metal/good conductor frame in a changing magnetic field

the electric field is not always zero. If the magnetic field changes steadily, a steady non-zero voltage builds up between A and B in accordance to faraday's induction law.
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  • 730
1 vote
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Metal/good conductor frame in a changing magnetic field

If the magnetic field into the screen is increasing, there will be a non-conservative electric field urging positive charge around the conductor from A towards B. Positive charge will then pile up on ...
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1 vote
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Why don't we know the electric potential at any point in a circuit, only the difference in the electric potential (voltage)?

In loop analysis where we apply Kirchhoff's voltage law (the algebraic sum of the potential differences around a loop equals zero) it is only necessary to consider potential differences. In doing node ...
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  • 56k
1 vote

Vector potential due to a spinning spherical shell with a non-uniform surface charge distribution

It seems very difficult to obtain analytically integrated formulas. By the way, in Landau and Lifshitz book (See PROBLEM 2 of page 125 in "Electrodynamics of Contnuous Media"), the ...
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