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The concept of de Broglie wavelength is incredibly fascinating as it bridges the worlds of classical and quantum physics. Named after French physicist Louis de Broglie, it proposes that all matter exhibits wave-like properties. This idea is central to the field of quantum mechanics. The de Broglie wavelength (\( \lambda \)) of a particle is given by the equation:\[\lambda = \frac{h}{mv}\]Here, \( h \) is Planck’s constant (\( 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \)), \( m \) is the mass of the particle, and \( v \) is its velocity.
This concept is particularly useful in describing particles at atomic and subatomic scales, where wave-particle duality is significant. For electrons, which are much smaller and have higher velocities, this wavelength can be calculated to determine interference and diffraction patterns. The smaller the particle, the more pronounced these effects become.

Diffraction minima occur when waves bend around an obstacle or pass through a slit and interfere with each other. This results in points of zero intensity, known as minima, where destructive interference occurs.
When an electron beam passes through a slit, it diffracts and spreads out, producing a pattern of dark and bright regions. The first diffraction minima can be found using the formula:\[a \sin(\theta) = m\lambda\]where \( a \) is the slit width, \( \theta \) is the angle of the minima, \( m \) is the order of minima, and \( \lambda \) is the wavelength of the particles. In our scenario, with the angle given as \( 11.5^{\circ} \) and the wavelength calculated using de Broglie's equation, we can determine the slit width.
This concept explains experiments like single-slit and double-slit experiments, which show light and electrons behaving as waves.

Relativistic effects come into play when objects move at speeds comparable to the speed of light (\( 3 \times 10^8 \text{ m/s} \)). At these speeds, classical mechanics fall short, and Einstein’s theory of relativity provides a more accurate description of physics.
For electrons accelerated through a potential difference, we calculate their speed to evaluate if relativistic effects are significant. We compare the electron's speed (\( v \)) with the speed of light, determining the ratio \( \frac{v}{c} \). If this ratio is greater than 0.1, relativistic corrections might be necessary.
In this case, the calculated speed was \( 5.93 \times 10^6 \text{ m/s} \), resulting in a very small fraction of the speed of light (\( \approx 0.02 \)). Hence, relativistic effects are negligible, allowing us to use classical formulas.

Calculating the speed of electrons after acceleration through a potential difference is crucial for many experiments and applications. The energy gained by each electron when accelerated through a voltage (\( V \)) can be determined using:\[ KE = eV = \frac{1}{2}mv^2 \]where \( e \) is the elementary charge, \( m \) is the electron mass, and \( v \) is the speed of the electrons.
Rearranging to find speed, the formula is:\[ v = \sqrt{\frac{2eV}{m}} \]By substituting the known values for an electron off a potential of 100 V, we found the speed to be \( 5.93 \times 10^6 \text{ m/s} \). This calculation is pivotal for applications involving electron beams, such as electron microscopy and particle acceleration experiments.