# Uniqueness Theorem, Concentric Equipotential Cylinders & Moving Charge

• rbrayana123
In summary, the conversation discusses the construction of a graph of current versus time for alpha particles emitted from two electrodes 2 mm apart in vacuum, with one particle moving directly towards the other electrode and the other moving at a 45 degree angle. It also considers the possibility of a cylindrical arrangement of electrodes and the shape of the resulting current pulse. The conversation also mentions the use of the Uniqueness Theorem and calculations for potential, electric field, and charge distribution in different scenarios. It ultimately concludes that the current pulse for the concentric cylinders would have a different shape than that of the plate capacitor due to the varying radius of the inner cylinder.
rbrayana123

## Homework Statement

Consider two electrodes 2 mm apart in vacuum connected by a short wire. An alpha particle of charge 2e is emitted by the left plate and travels directly towards the right plate with constant speed 106 m/s and stops in this plate. Make a quantitative graph of the current in the connecting wire, plotting current against time. Do the same for an alpha particle that crosses the gap moving with the same speed but at an angle of 45°. Suppose we had a cylindrical arrangement of electrodes with alpha particles being emitted from a thin wire on the axis of a small cylindrical electrode. Would the current pulse have the same shape?

## Homework Equations

Uniqueness Theorem
For a cylinder, E = 2kλ/r

## The Attempt at a Solution

d = 2 * 10-3 m
v = 106 m/s

Let Plate 1 be positioned at x = 0; point charge at x = a; and Plate 2 at x = d.
Also, Plate 3 will be positioned at x = a in different scenario;
E1 is the electric field between Plates 1 & 3;
E2 is the electric field between Plates 2 & 3.

Application of Uniqueness Theorem: If I find a solution for specific charges Q1 on Plate 1 & Q2 on Plate 2 given a point charge located at a, then it is THE solution. (Did I miss any points? I'm still a little iffy on this.)

Placing Plate 3 of total charge Q parallel to the other two plates and in the plane of x = a will distribute the same charges Q1 & Q2 as the a single point charge at x = a.

E1 = σ1o
E2 = -σ2o

Because they are connected by wires, the potentials are equal.

σ1o(-a) = -σ2o(d-a)
σ2 = σ1a/(d-a)

Q2 = Q1a/(d-a)
Q1 + Q2 = -Q

Q1 = -Q$\frac{d-a}{d}$
Q2 = -Q$\frac{a}{d}$

Now, considering that there is a moving charge, a = vt, Q = 2e
Q2 = -2e$\frac{vt}{d}$
I = dQ/dt = -2ev/d

The graph of current versus time for a plate capacitor is constant. Now, if the charge were moving at a 45 degree angle, it's just a vector consideration.
I = dQ/dt = -2ev/d cos(45°) = -sqrt(2)ev/d

Here's the hard part: The concentric cylinders

Let the inner cylinder be of radius a with total charge Q1; and the outer cylinder be of radius b with total charge Q2.

Application of Uniqueness Theorem: If I find a solution for specific charges Q1 on Cylinder 1 & Q2 on Cylinder 2 given a point charge located at vt, then it is THE solution. Now, let me consider Cylinder 3 of radius vt with total charge Q to satisfy the above conditions.

E1 = 2kλ1/r
E2 = 2kλ2/r

Because they are connected by wires, the potentials are equal:

2kλ1ln(vt/a) = 2kλ2ln(b/vt)

Q2 = Q1$\frac{ln(vt/a)}{ln(b/vt)}$
Q1 + Q2 = -2e
Q1 (1 + $\frac{ln(vt/a)}{ln(b/vt)}$) = -2e
Q1 $\frac{ln(vt/a) + ln(b/vt)}{ln(b/vt)}$ = -2e
Q1 $\frac{ln(b/a)}{ln(b/vt)}$ = -2e
Q1 = -2e $\frac{ln(b/vt)}{ln(b/a)}$

I = dQ1/dt = 2e$\frac{1}{tln(b/a)}$

^This seems really off...

The current pulse for the concentric cylinders will have a different shape than the plate capacitor due to the varying radius of the inner cylinder.

## 1. What is the Uniqueness Theorem?

The Uniqueness Theorem is a principle in electrostatics that states that the solution to a boundary value problem is unique, meaning there can only be one solution that satisfies all the given conditions. This theorem is useful in solving problems involving electric fields and potential.

## 2. How are concentric equipotential cylinders related to the Uniqueness Theorem?

Concentric equipotential cylinders are a special case of the Uniqueness Theorem, where the electric field inside a set of nested cylinders is the same for all points on the same cylinder. This means that the potential difference between any two points on the same cylinder is zero, making the cylinders equipotential surfaces.

## 3. What is the significance of concentric equipotential cylinders in electrostatics?

In electrostatics, concentric equipotential cylinders are used to model the electric field and potential around a charged object. They help simplify calculations and provide a visual representation of the electric field lines and equipotential surfaces. They are also useful in understanding the behavior of electric fields in conductors.

## 4. How does the Uniqueness Theorem apply to moving charges?

The Uniqueness Theorem also applies to moving charges, as it states that the solution to a boundary value problem is unique, regardless of the motion of the charges. This means that the electric fields and potentials around moving charges can be calculated using the same principles as stationary charges.

## 5. Can the Uniqueness Theorem be applied to other physical phenomena?

Yes, the Uniqueness Theorem has applications in various fields of physics, such as magnetostatics and fluid dynamics. It states that the solution to a boundary value problem is unique, regardless of the physical phenomenon being studied. This makes it a powerful tool in solving complex problems in different areas of physics.

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