The wave like character is significant and measurable for

Options

A. macroscopic objects only
B. sub-atomic particles
C. both macroscopic objects and sub-atomic particles
D. heavy and more energetic particles

Answer

To solve the question regarding the significance of wave-like character for different types of objects, we will analyze the de Broglie wavelength for both macroscopic and subatomic particles.

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (λ) is given by the formula:
\[ λ = \frac{h}{mv} \]
where:
- \( h \) is Planck's constant (\( 6.63 \times 10^{-34} \, \text{Js} \)),
- \( m \) is the mass of the object,
- \( v \) is the velocity of the object.

### Step 2: Calculate the de Broglie wavelength for a macroscopic object
Let's consider a macroscopic object, such as a ball with a mass of 46 grams (which is \( 46 \times 10^{-3} \, \text{kg} \)) moving at a speed of 30 m/s.

Substituting the values into the formula:
\[ λ = \frac{6.63 \times 10^{-34}}{(46 \times 10^{-3}) \times 30} \]
Calculating the denominator:
\[ (46 \times 10^{-3}) \times 30 = 1.38 \, \text{kg m/s} \]
Now substituting back:
\[ λ = \frac{6.63 \times 10^{-34}}{1.38} \approx 4.8 \times 10^{-34} \, \text{m} \]

### Step 3: Calculate the de Broglie wavelength for a subatomic particle
Now, let's consider an electron with a mass of \( 9.1 \times 10^{-31} \, \text{kg} \) moving at a speed of \( 10^7 \, \text{m/s} \).

Using the same formula:
\[ λ = \frac{6.63 \times 10^{-34}}{(9.1 \times 10^{-31}) \times (10^7)} \]
Calculating the denominator:
\[ (9.1 \times 10^{-31}) \times (10^7) = 9.1 \times 10^{-24} \, \text{kg m/s} \]
Now substituting back:
\[ λ = \frac{6.63 \times 10^{-34}}{9.1 \times 10^{-24}} \approx 7.12 \times 10^{-11} \, \text{m} \]

### Step 4: Compare the wavelengths
Now we compare the two wavelengths:
- For the macroscopic object (ball): \( 4.8 \times 10^{-34} \, \text{m} \)
- For the subatomic particle (electron): \( 7.12 \times 10^{-11} \, \text{m} \)

Clearly, \( 7.12 \times 10^{-11} \, \text{m} \) is significantly larger than \( 4.8 \times 10^{-34} \, \text{m} \).

### Conclusion
The wave-like character is significant and measurable for subatomic particles, as indicated by the larger de Broglie wavelength calculated for the electron compared to the macroscopic object.

### Final Answer
The wave-like character is significant and measurable for **subatomic particles**.

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