Q.What physical quantity is the same for X-rays of wavelength $10^{-10}\ \text{m}$, red light of wavelength $6800\ \text{Å}$ and radiowaves of wavelength $500\ \text{m}$?

The key idea is that all electromagnetic waves travel at the same speed in vacuum, regardless of wavelength or frequency.

Reasoning:

1. X-rays, red light, and radiowaves are all forms of electromagnetic radiation.
2. In vacuum, every electromagnetic wave propagates with the speed of light, $c = 3 \times 10^8\ \text{m/s}$.
3. The wavelength changes across the spectrum, but the speed in vacuum remains invariant.

The question lists three very different electromagnetic waves: X-rays ($10^{-10}\ \text{m}$), red light ($6800\ \text{Å}$), and radiowaves ($500\ \text{m}$). The wavelengths span an enormous range — from atomic scales to hundreds of metres. Yet all three are fundamentally the same kind of phenomenon: electromagnetic radiation.

The key insight is that all electromagnetic waves travel at the same speed in vacuum. This speed, denoted $c$, is a universal constant of nature. It does not depend on wavelength, frequency, or intensity. Whether it's gamma rays or long radio waves, in empty space they all move at $c = 3 \times 10^8\ \text{m/s}$.

Let's verify this step by step.

1. Identify the nature of the waves. X-rays, visible light, and radiowaves are all electromagnetic waves. They differ only in wavelength (or frequency), but their physical origin is the same — oscillating electric and magnetic fields propagating through space.

2. Recall the universal constant for electromagnetic waves. In vacuum, Maxwell's equations predict that all electromagnetic waves travel at a speed

This is a fundamental constant, independent of the wave's wavelength or frequency.

3. Check the relationship between speed, frequency, and wavelength. For any wave,

For electromagnetic waves in vacuum, $v = c$, so

If you know the wavelength, you can find the frequency, but the speed $c$ remains unchanged.

4. Apply to the given waves.
   • X-rays: $\lambda = 10^{-10}\ \text{m}$ → $f = c/\lambda \approx 3 \times 10^{18}\ \text{Hz}$
   • Red light: $\lambda = 6800\ \text{Å} = 6800 \times 10^{-10}\ \text{m} = 6.8 \times 10^{-7}\ \text{m}$ → $f \approx 4.4 \times 10^{14}\ \text{Hz}$
   • Radiowaves: $\lambda = 500\ \text{m}$ → $f \approx 6 \times 10^5\ \text{Hz}$ The frequencies are wildly different, but the speed is the same $c$ for all three.

Method: Invariance of Speed of Light in Vacuum

Concept: All electromagnetic waves — regardless of wavelength or frequency — travel at the same speed in vacuum.

Steps

1. Identify the common medium

   X-rays, red light, and radiowaves are all electromagnetic waves. Unless specified otherwise, they are assumed to travel in vacuum (or air, which is nearly the same for this purpose).

2. Recall the universal constant

   In vacuum, every electromagnetic wave propagates at the speed of light:

3. Check if any wave is slowed

   • X-rays ($\lambda = 10^{-10}\ \text{m}$) — travel at $c$ in vacuum.
   • Red light ($\lambda = 6800\ \text{Å} = 6.8 \times 10^{-7}\ \text{m}$) — travel at $c$ in vacuum.
   • Radiowaves ($\lambda = 500\ \text{m}$) — travel at $c$ in vacuum.

   No dependence on wavelength — the speed is identical for all.

4. State the answer

   The physical quantity that is the same for all three is the speed in vacuum:

Why this works (exam tip)

• In vacuum, $c = \nu \lambda$ is constant — if wavelength changes, frequency adjusts to keep product constant.
• In a medium, speed changes (e.g., glass slows light), but the frequency remains the same across media — not the speed. Here, since no medium is mentioned, assume vacuum.

Here are the most common mistakes students make on this question, along with the reasoning to avoid them.

Mistake 1: Thinking "Wavelength" or "Frequency" is the Same

• The error: Students see three different wavelengths ($10^{-10}\ \text{m}$, $6800\ \text{Å}$, $500\ \text{m}$) and assume the question is asking for a quantity that is numerically equal for all three. They then try to convert units and compare.
• Why it's wrong: The wavelengths are vastly different. The frequency is also different (since $f = c/\lambda$). The question asks for a physical quantity that is identical for all electromagnetic waves, not a numerical value that matches.
• How to avoid: Read the question carefully. It asks: "What physical quantity is the same for X-rays, red light, and radiowaves?" The answer is a property common to all electromagnetic radiation, not a calculation.

Mistake 2: Confusing "Speed" with "Velocity" or Ignoring the Medium

• The error: Some students write "velocity" or "speed in a vacuum" but then get confused if the problem implies air or another medium. They might think speed changes with wavelength.
• Why it's wrong: In a vacuum, all electromagnetic waves travel at the same speed: $c = 3 \times 10^8\ \text{m/s}$. In air, the speed is slightly less but still essentially the same for all these waves. The key is that speed does not depend on wavelength or frequency in a given medium.
• How to avoid: Memorise the fundamental property: In a vacuum, the speed of all electromagnetic waves is constant ($c$). If the medium is not specified, assume vacuum or air, where the speed is the same for all.

Mistake 3: Giving "Frequency" or "Wavelength" as the Answer

• The error: A student might calculate the frequency for one wave and, seeing it's a large number, think it's the same for all. Or they might think "wavelength" is a universal property.
• Why it's wrong: Frequency and wavelength are inversely proportional ($f = c/\lambda$). Since the wavelengths given are different, the frequencies are also different. For example:
  • X-ray: $\lambda = 10^{-10}\ \text{m} \implies f \approx 3 \times 10^{18}\ \text{Hz}$
  • Red light: $\lambda = 6800\ \text{Å} = 6.8 \times 10^{-7}\ \text{m} \implies f \approx 4.4 \times 10^{14}\ \text{Hz}$
  • Radiowave: $\lambda = 500\ \text{m} \implies f \approx 6 \times 10^{5}\ \text{Hz}$ These are clearly not the same.
• How to avoid: Always check: if the wavelengths are different, the frequencies must be different (in the same medium). The only quantity that remains constant across the entire electromagnetic spectrum is speed in vacuum.

Mistake 4: Forgetting the Unit Conversion for Wavelength

• The error: A student tries to compare the wavelengths numerically without converting all to the same unit (e.g., leaving $6800\ \text{Å}$ as is and comparing it to $10^{-10}\ \text{m}$).
• Why it's wrong: You cannot compare $6800$ with $10^{-10}$ directly. $1\ \text{Å} = 10^{-10}\ \text{m}$, so $6800\ \text{Å} = 6.8 \times 10^{-7}\ \text{m}$. This is still vastly different from $10^{-10}\ \text{m}$ and $500\ \text{m}$.
• How to avoid: Always convert all quantities to SI units (metres) before any comparison or calculation. But in this question, you don't even need to calculate — just recognise the concept.

Mistake 5: Overcomplicating with "Energy" or "Momentum"

• The error: Some students think about photon energy ($E = hf$) or momentum ($p = h/\lambda$) and try to see if those are the same.
• Why it's wrong: Energy and momentum depend on frequency or wavelength. Since the frequencies are different, the energies and momenta are also different.
• How to avoid: Remember that only speed in vacuum is invariant for all electromagnetic waves. Energy, momentum, frequency, and wavelength all change across the spectrum.

✓ The Correct Answer

The physical quantity that is the same for X-rays, red light, and radiowaves is:

Speed in vacuum (or in air).

All electromagnetic waves travel at $c = 3 \times 10^8\ \text{m/s}$ in vacuum.

Final Answer: $\boxed{\text{Speed}}$