Difference in potential energy of two charge configurations

theideasmith
Messages
1
Reaction score
0
Chapter 24, Question 61

Given two configurations, ##C_1##, ##C_2## of ##N## point charges each, determine the smallest value of ##N## s.t. ##V_1>V_2##.

##C_1##:

##N## point charges are uniformly distributed on a ring s.t. the distance between adjacent electrons is constant

##C_2##:

##N-1## point charges are uniformly distributed on a ring s.t. the distance between adjacent electrons is constant and one charge is placed in the center of the ring.

Approach I

1. If we consider a gaussian surface inside the ring, ##E=0##. We know that the voltage at the center of the ring is $$V_\text{center}=\frac{Ne}{r}$$ and furthermore, because $V=\int E\circ ds$, $$V_\text{inside} = V_\text{center}$$
2. From this previous result, $$U_1 = eV_\text{center} = \frac{N(N-1)e^2}{r}$$
3. ##C_2##
the configuration potential without the center electron is $$(1/2)(N-1)(N-2)\frac{e}{r}$$ The center electron adds ##(N-1)\frac{e}{r}## yielding $$U_2 = (1/2)(N-1)(N-2)\frac{e}{r} + (N-1)\frac{e}{r}$$
4. Let ##k = \frac{e}{r}##, and, setting ##(N^2-N)k = (N^2-3N+2)k+(2N-2)k##
$$\implies 0=0$$

Approach II

Let ##N## charges be arranged along a circle with radius ##R##. The position of an arbitrary particle at an angle ##\theta## relative to the positive direction of the x-axis is ##\vec{P}(\theta) = (R\cos\theta, R\sin\theta)##. Pick one charge located at angle $\theta = \theta_i$ and another particle located at ##\theta = \theta_j## relative to the positive direction of the x-axis. The distance between the two particles ##\vec{r}## is

$$
\begin{align}
\vec{r} &= \|\vec{P}(\theta_j) - \vec{P}(\theta_i)\| \\
&= \|(R\cos\theta_j, R\sin\theta_j) - (R\cos\theta_i, R\sin\theta_i) \\
&= \|R(\cos\theta_j-\cos\theta_i, \sin\theta_j - \sin\theta_j)\|\\
&= R\|(\cos\theta_j-\cos\theta_i, \sin\theta_j - \sin\theta_j)\| \\
&= R \sqrt{
\cos^2\theta_j -2 \cos\theta_j\cos\theta_i + \cos^2\theta_i \\
+ \sin^2\theta_j -2 \sin\theta_j \sin\theta_i + \sin^2\theta_i
} \\
&= R\sqrt{
1-\sin^2\theta_j +1-\sin^2\theta_i + \sin^2\theta_j + \sin^2\theta_i \\
-2(\cos\theta_j\cos\theta_i+ \sin\theta_j \sin\theta_i)
}\\
&= R\sqrt{
2-2(\cos\theta_j \cos\theta_i + \sin\theta_j \sin\theta_i)
}\\
&=R\sqrt{2(1-\cos\theta_j \cos\theta_i - \sin\theta_j \sin\theta_i) } \\
& = R\sqrt{2(1-\cos(\theta_j + \theta_i))}\\
\therefore \vec{r} &= R\sqrt{2(1-\cos(\theta_j + \theta_i))}
\end{align}
$$

Where

$$
\theta_k = k\Delta \theta \\
\Delta\theta = \frac{2\pi}{N}
$$

We get

$$
\begin{aligned}
\vec{r} =R\sqrt{2}\sqrt{1-\cos\left((i+j)\frac{2\pi}{N}\right)}
\end{aligned}
$$

With this expression for ##\vec{r}##, we can write the net potential energy for particle ##j## along the circle with the equation assuming all particles have charges ##q##

$$
\begin{align}
U_\text{particle i} & = \sum_{j=1, i \ne j}^{N}
{
\frac{q^2}{4\pi\epsilon_0R\sqrt{2(1-\cos(\theta_i + \theta_j))}}}\\
& = \frac{q^2}{4\pi\epsilon_0R\sqrt{2}}\sum_{j=1, i \ne j}^{N}
{
\frac{1}{\sqrt{1-\cos(\theta_i + \theta_j)}}} \\
& = \frac{q^2}{4\pi\epsilon_0R\sqrt{2}}\sum_{j=2}^{N}
{
\frac{1}{\sqrt{1-\cos(\theta_i + \theta_j)}}}
\end{align}
$$

we set ##j_\text{initial} =2## which is equivalent to the conditions ##j=1, j\ne i##
For concision, let

$$
L = \frac{q^2}{4\pi\epsilon_0R}
$$

Then net potential energy can be expressed as

$$
\begin{align}
U(n)
& = \frac{1}{2}\sum_i^{n}U_i \\
& = \frac{L}{2} \sum_{i=1, j=2}^{n,n}
{
\frac{1}{\sqrt{2-2\cos(\Delta \theta(i+j-2))}}
} \\
&= \frac{nq^2}{8\pi\epsilon_0R}
\sum_{j=1}^{n-1}{
\frac{1}{\sqrt{2}\sqrt{1-\cos(j\Delta\theta))}}
}\\
&= \frac{nq^2}{8\pi\epsilon_0R}
\sum_{j=1}^{n-1}{
\frac{1}{\sqrt{2}\sqrt{1-\cos(j\frac{2\pi}{N}))}}
}\\
&= \frac{nq^2}{8\pi\epsilon_0R}
\sum_{j=1}^{n-1}{
\frac{1}{4\sin(\frac{j\pi}{N})}
}\\
&= \frac{nq^2}{32\pi\epsilon_0R}
\sum_{j=1}^{n-1}{
\csc\left(j\frac{\pi}{N}\right)
} \\
\end{align}
$$

For two configurations with ##N## charges we define the potential energies:

$$
U_1 = U(N) \\
U_2 = U(N-1) + (N-1)L
$$

where the second term in the definition of ##U_2## determines the potential for the lone particle in the center.

Now we solve for the ##N## at which ##U_1 = U_2##

$$
\begin{aligned}
U_1 - U_2& = \Delta U \\
&=N\frac{q^2}{32\pi\epsilon_0R}
\sum_{j=1}^{n-1}{
\csc\left(j\frac{\pi}{N}\right)
}\\
&+(1-N)\frac{q^2}{32\pi\epsilon_0R}
\sum_{j=1}^{n-2}{
\csc\left(j\frac{\pi}{N-1}\right)
}\\
& +(1-N)L
\end{aligned}

$$

Which is really difficult to solve directly.
 
Physics news on Phys.org
theideasmith said:
Given two configurations, ##C_1##, ##C_2## of N point charges each, determine the smallest value of ##N## s.t. ##V_1>V_2##.
Are ##V_1## and ##V_2## the electrostatic potential energies associated with each distribution?
theideasmith said:
If we consider a gaussian surface inside the ring, ##E=0##
How do you figure? The electric flux out of the Gaussian surface is zero, but this doesn't mean that the electric field everywhere must be zero as a result. For example, what if the configuration has N = 2?
2charges.png
 
theideasmith said:
Which is really difficult to solve directly.

After fixing up my algebra, I ended up with the equation

$$\begin{aligned}
U_2-U_1 &= 0 \\
&= L\frac{N-1}{2}\sum_{j=1}^{N-2}{\csc\left(\frac{j\pi}{N-1}\right)}
+ L(N-1)2
- L\frac{N}{2}\sum_{j=1}^{N-1}{\csc\left(\frac{j\pi}{N}\right)} \\
&= L\left(\frac{N-1}{2}\sum_{j=1}^{N-2}{\csc\left(\frac{j\pi}{N-1}\right)}
- \frac{N}{2}\sum_{j=1}^{N-1}{\csc\left(\frac{j\pi}{N}\right)}
+ 2(N-1)
\right)
\end{aligned}$$

and numerically solved for ##N## by plotting the series for ##U_1## and ##U_2##.
MDdbAwt.png

I got ##N=12##.

Lesson: Sometimes brute force is necessary.
Full solution: http://theideasmith.github.io/2017/02/14/Point-Charge-Configuration.html
 
To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.
Back
Top