Table 1 shows results of an experiment to investigate how the de Broglie wavelength $ \lambda $ of an electron varies with its velocity $ v $. | v / 10^7 m s^-1 | $ \lambda / 10^{-11} $ | |----------------|--------------------| | 1.5 | 4.9 | | 2.5 | 2.9 | | 3.5 | 2.1 | Show that the data in Table 1 are consistent with the relationship $ \lambda \propto \frac{1}{v} $ [2 marks] Calculate a value for the Planck constant suggested by the data in Table 1. [2 marks] Figure 2 shows the side view of an electron diffraction tube used to demonstrate the wave properties of an electron. An electron beam is incident on a thin graphite target that behaves like the slits in a diffraction grating experiment. After passing through the graphite target the electrons strike a fluorescent screen. Figure 3 shows the appearance of the fluorescent screen when the electrons are incident on it. Explain how the pattern produced on the screen supports the idea that the electron beam is behaving as a wave rather than as a stream of particles. [3 marks] Explain how the emission of light from the fluorescent screen shows that the electrons incident on it are behaving as particles. [3 marks]

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Table 1 shows results of an experiment to investigate how the de Broglie wavelength $ \lambda $ of an electron varies with its velocity $ v $ - AQA - A-Level Physics - Question 2 - 2018 - Paper 1

Question 2

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Table 1 shows results of an experiment to investigate how the de Broglie wavelength $ \lambda $ of an electron varies with its velocity $ v $. | v / 10^7 m s^-1... show full transcript

Worked Solution & Example Answer:Table 1 shows results of an experiment to investigate how the de Broglie wavelength $ \lambda $ of an electron varies with its velocity $ v $ - AQA - A-Level Physics - Question 2 - 2018 - Paper 1

Step 1

Show that the data in Table 1 are consistent with the relationship $ \lambda \propto \frac{1}{v} $

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Answer

To show that the data from Table 1 is consistent with ( \lambda \propto \frac{1}{v} ), we can analyze the relationship between the wavelength ( \lambda ) and the velocity ( v ) for the values provided:

For ( v = 1.5 \times 10^7 \text{ m s}^{-1} ), ( \lambda = 4.9 \times 10^{-11} \text{ m} )
For ( v = 2.5 \times 10^7 \text{ m s}^{-1} ), ( \lambda = 2.9 \times 10^{-11} \text{ m} )
For ( v = 3.5 \times 10^7 \text{ m s}^{-1} ), ( \lambda = 2.1 \times 10^{-11} \text{ m} )

To check the proportionality, we calculate the ratio ( v \lambda ) for each entry:

For 1.5: ( v \ imes \lambda = (1.5 \times 10^7) \times (4.9 \times 10^{-11}) = 0.735 )
For 2.5: ( v \ imes \lambda = (2.5 \times 10^7) \times (2.9 \times 10^{-11}) = 0.725 )
For 3.5: ( v \ imes \lambda = (3.5 \times 10^7) \times (2.1 \times 10^{-11}) = 0.735 )

The values are consistently around 0.735, supporting the relationship ( \lambda \propto \frac{1}{v} ) since as velocity increases, wavelength decreases.

Step 2

Calculate a value for the Planck constant suggested by the data in Table 1.

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Answer

The de Broglie wavelength is given by the formula:

$\lambda = \frac{h}{mv}$

Where ( h ) is the Planck constant, ( m ) is the mass of the electron (approximately ( 9.11 \times 10^{-31} \text{ kg} )), and ( v ) is the velocity.

We can rearrange this to calculate ( h ):

$h = \lambda mv$

We can use the first row of data:

( \lambda = 4.9 \times 10^{-11} \text{ m} )
( v = 1.5 \times 10^7 \text{ m s}^{-1} )

Now, substituting the values into the formula:

$h = (4.9 \times 10^{-11}) \times (9.11 \times 10^{-31}) \times (1.5 \times 10^7)$

Calculating this gives:

$h \approx 7.00 \times 10^{-34} \text{ Js}$

Thus, the calculated value for the Planck constant suggested by the data in Table 1 is approximately ( 7.00 \times 10^{-34} \text{ Js} ).

Step 3

Explain how the pattern produced on the screen supports the idea that the electron beam is behaving as a wave rather than as a stream of particles.

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Answer

The pattern produced on the fluorescent screen indicates interference, which is a hallmark of wave behavior. When electrons pass through the thin graphite target, they diffract and overlap, creating a series of bright and dark fringes on the screen. This pattern results from constructive and destructive interference, which occurs when waves overlap:

Constructive interference occurs where the waves are in phase, resulting in increased brightness.
Destructive interference occurs where the waves are out of phase, leading to darkness.

This observed pattern is fundamentally different from what one would expect if electrons were merely behaving as particles; a particle stream would produce two distinct spots corresponding to the slits rather than a continuous pattern of interference.

Step 4

Explain how the emission of light from the fluorescent screen shows that the electrons incident on it are behaving as particles.

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Answer

When electrons strike the fluorescent screen, they impart energy to the atoms in the screen's material, causing those atoms to emit photons of light. This energy transfer is an indication of particle behavior because:

Each electron can be viewed as a discrete packet of energy, which interacts with the material at a specific location.
The emission of light indicates that the electrons have transferred kinetic energy to the screen, supporting the concept of electrons behaving as particles that can collide with other particles (atoms in the screen).

Therefore, while the interference pattern illustrates wave-like behavior, the emission of light confirms that the electrons are also behaving like particles during their interaction with the screen.

Table 1 shows results of an experiment to investigate how the de Broglie wavelength \( \lambda \) of an electron varies with its velocity \( v \) - AQA - A-Level Physics - Question 2 - 2018 - Paper 1

Worked Solution & Example Answer:Table 1 shows results of an experiment to investigate how the de Broglie wavelength \( \lambda \) of an electron varies with its velocity \( v \) - AQA - A-Level Physics - Question 2 - 2018 - Paper 1

Show that the data in Table 1 are consistent with the relationship \( \lambda \propto \frac{1}{v} \)

Calculate a value for the Planck constant suggested by the data in Table 1.

Explain how the pattern produced on the screen supports the idea that the electron beam is behaving as a wave rather than as a stream of particles.

Explain how the emission of light from the fluorescent screen shows that the electrons incident on it are behaving as particles.

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