A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain

Q: Why do we use dimensional analysis here instead of deriving the formula from scratch?

Deriving the plasma frequency requires setting up the equation of motion for the electron cloud under a restoring electric field, which can be time-consuming. Dimensional analysis is a powerful shortcut in competitive exams to quickly identify the correct expression among given options.

Q: What is the physical meaning of plasma frequency?

It is the natural frequency at which the electron cloud in a plasma oscillates back and forth relative to the heavy, stationary positive ions when displaced.

Q: Why does reflection occur when $\omega$ approaches $\omega_p$?

When the frequency of the incident electromagnetic wave matches the plasma frequency, the free electrons resonate. They absorb the wave's energy and re-radiate it, effectively reflecting the wave. This is why metals are shiny!

Q: How did we get the dimension of $\varepsilon_0$?

You can derive it from Coulomb's law: $F = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2}$. Rearranging for $\varepsilon_0$ gives $[\varepsilon_0] = \frac{[Q^2]}{[F][L^2]} = \frac{[\text{A}^2\text{T}^2]}{[\text{MLT}^{-2}][\text{L}^2]} = [\text{M}^{-1}\text{L}^{-3}\text{T}^4\text{A}^2]$.

Q: Why did we approximate $10.24$ to $10$ in the calculation?

In physics estimation problems, especially when the options are widely spaced (like 200, 300, 600, 800), making small approximations (like $10.24 \approx 10$) drastically speeds up the calculation without changing the final closest option.

Question

Passage:
A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let $N$ be the number density of free electrons, each of mass $m$. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency $\omega_p$, which is called the plasma frequency.

To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency $\omega$, where a part of the energy is absorbed and a part of it is reflected. As $\omega$ approaches $\omega_p$, all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.

Question 1:
Taking the electronic charge as $e$ and the permittivity as $\varepsilon_0$, use dimensional analysis to determine the correct expression for $\omega_p$.
(A) $\sqrt{\frac{Ne}{m\varepsilon_0}}$
(B) $\sqrt{\frac{m\varepsilon_0}{Ne}}$
(C) $\sqrt{\frac{Ne^2}{m\varepsilon_0}}$
(D) $\sqrt{\frac{m\varepsilon_0}{Ne^2}}$

Question 2:
Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons $N \approx 4 	imes 10^{27} 	ext{ m}^{-3}$. Take $\varepsilon_0 \approx 10^{-11}$ and $m \approx 10^{-30}$, where these quantities are in proper SI units.
(A) $800 	ext{ nm}$
(B) $600 	ext{ nm}$
(C) $300 	ext{ nm}$
(D) $200 	ext{ nm}$

Sigma Percentile · Accepted Answer

### The Physics of Plasma Oscillations

Imagine a neutral plasma, a dense collection of an equal number of electrons and positive ions.

When the free electrons are displaced from the heavy, stationary positive ions by an external electric field, a restoring force acts on them.

If the electric field is suddenly removed, the electrons don't just return to their original positions; they overshoot and begin to oscillate.

This natural oscillation occurs at a specific frequency known as the **plasma frequency**, denoted by $\omega_p$.

### Dimensional Analysis: Unlocking the Formula

Our first challenge is to determine the correct mathematical expression for this plasma frequency using dimensional analysis.

Instead of deriving the complex equations of motion, we can use the dimensions of the given physical quantities to find the right formula.

Let's list the dimensions of the variables involved:
- Number density of electrons, $[N] = 	ext{L}^{-3}$
- Elementary charge, $[e] = 	ext{AT}$
- Mass of an electron, $[m] = 	ext{M}$
- Permittivity of free space, $[\varepsilon_0] = 	ext{M}^{-1}	ext{L}^{-3}	ext{T}^4	ext{A}^2$

We know that angular frequency $\omega_p$ has the dimension of inverse time, $[	ext{T}^{-1}]$.

By testing the given options, we evaluate the term $\frac{Ne^2}{m\varepsilon_0}$.

Substituting the dimensions, we get:
$$ \left[ \frac{Ne^2}{m\varepsilon_0} ight] = \frac{(	ext{L}^{-3})(	ext{A}^2	ext{T}^2)}{(	ext{M})(	ext{M}^{-1}	ext{L}^{-3}	ext{T}^4	ext{A}^2)} = 	ext{T}^{-2} $$

Notice how the mass ($	ext{M}$) and length ($	ext{L}$) dimensions perfectly cancel out, leaving us with inverse time squared ($	ext{T}^{-2}$).

Taking the square root of this term gives us the dimension of inverse time, $[	ext{T}^{-1}]$, which perfectly matches the angular frequency.

Therefore, the correct expression is:
$$ \omega_p = \sqrt{\frac{Ne^2}{m\varepsilon_0}} $$

### The Resonance Condition

Now, let's tackle the second part of the problem.

We need to estimate the wavelength at which plasma reflection occurs for a specific metal.

Plasma reflection happens when the frequency of the incident electromagnetic wave ($\omega$) matches the natural plasma frequency ($\omega_p$).

At this resonance condition, the free electrons absorb the wave's energy and re-radiate it, effectively reflecting the wave.

We can relate the plasma frequency to the wavelength ($\lambda$) using the wave equation:
$$ \omega_p = \frac{2\pi c}{\lambda} $$

Rearranging this formula to solve for wavelength, we get:
$$ \lambda = \frac{2\pi c}{\omega_p} = 2\pi c \sqrt{\frac{m\varepsilon_0}{Ne^2}} $$

### Calculating the Wavelength

With our master equation ready, it's time to substitute the given SI values.

We are given:
- $c = 3 	imes 10^8 	ext{ m/s}$
- $N \approx 4 	imes 10^{27} 	ext{ m}^{-3}$
- $\varepsilon_0 \approx 10^{-11} 	ext{ F/m}$
- $m \approx 10^{-30} 	ext{ kg}$
- $e \approx 1.6 	imes 10^{-19} 	ext{ C}$

Let's first calculate the term inside the square root.

The numerator is:
$$ m\varepsilon_0 = (10^{-30})(10^{-11}) = 10^{-41} $$

The denominator is:
$$ Ne^2 = (4 	imes 10^{27})(1.6 	imes 10^{-19})^2 \approx 10.24 	imes 10^{-11} $$

Dividing the numerator by the denominator gives:
$$ \frac{m\varepsilon_0}{Ne^2} = \frac{10^{-41}}{10.24 	imes 10^{-11}} \approx 9.76 	imes 10^{-32} $$

Taking the square root of this value yields approximately $3.125 	imes 10^{-16} 	ext{ s}$.

Finally, we multiply this result by $2\pi c$:
$$ \lambda = 2 \pi (3 	imes 10^8) (3.125 	imes 10^{-16}) $$

$$ \lambda \approx 589 	imes 10^{-9} 	ext{ m} $$

Rounding to the nearest given option, we find that the plasma reflection occurs at a wavelength of **$600 	ext{ nm}$**.

Animated Solution for Physics - Physics and Measurement: Comprehension Passage

Taking the electronic charge as $e$ and the permittivity as $ε_{0}$ , use dimensional analysis to determine the correct expression for $ω_{p}$ .

Select Answer:

Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons $N \approx 4 \times 1 0^{27} m^{- 3}$ . Take $ε_{0} \approx 1 0^{- 11}$ and $m \approx 1 0^{- 30}$ , where these quantities are in proper SI units.

Select Answer:

Visualized Solution

$Dimensional Analysis$

$Dimensions of Variables$

$Testing the Options$

$Final Expression for ω_{p}$

$Wavelength Relation$

$Substituting Values$

$Calculating the Terms$

$Evaluating the Square Root$

$Final Wavelength$

The Sigma Insight: Dimensional Analysis

The Physics of Plasma Oscillations

Dimensional Analysis

The Resonance Condition

Calculating the Wavelength

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Animated Solution for Physics - Physics and Measurement: Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Taking the electronic charge as e and the permittivity as ε0​, use dimensional analysis to determine the correct expression for ωp​.

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Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Dimensional Analysis

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Dimensions of Variables

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Testing the Options

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Expression for ωp​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Wavelength Relation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Substituting Values

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Calculating the Terms

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluating the Square Root

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Wavelength

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The Physics of Plasma Oscillations

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