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When electrons move in a television set, they are sped up or accelerated using a potential difference. Imagine potential difference as a means of giving energy to the electrons, which propels them forward. This is similar to giving a ball a push to roll it down a hill. The potential difference provides the energy needed to increase the speed of the electrons. In this exercise, the given potential difference is 25.0 kV, which is equivalent to 25,000 volts.
The potential difference creates an electric field that pushes the electrons, allowing them to gain speed as they travel through its influence. Understanding electron acceleration in this way helps to conceptualize how electrons acquire the necessary speed to reach the television screen and create an image.

The potential difference is crucial in calculating the kinetic energy gained by the electrons. Kinetic energy is determined using the formula: \[ KE = eV \] where \( e \) is the electron charge \( (1.6 \times 10^{-19} \, \text{C}) \) and \( V \) is the potential difference (25,000 V in this scenario). This equation shows that kinetic energy is directly related to the amount of work done on electrons by the potential difference.

• High potential difference = More kinetic energy
• Low potential difference = Less kinetic energy

For our exercise, multiplying the electron charge by the potential difference gives a kinetic energy of \( 4.0 \times 10^{-15} \, \text{J} \). This value is essential for further calculations in determining the de Broglie wavelength.

Understanding momentum is crucial for computing the de Broglie wavelength. Momentum, denoted as \( p \), connects with kinetic energy through the equation: \[ KE = \frac{p^2}{2m} \] Here, \( m \) is the mass of an electron \( (9.11 \times 10^{-31} \, \text{kg}) \). Rearranging this equation allows us to solve for momentum: \[ p = \sqrt{2mKE} \]
By substituting the known values for electron mass and kinetic energy, we can calculate the momentum of the electron. In this case, the result is \( 6.02 \times 10^{-24} \, \text{kg} \, \text{m/s} \). Calculating momentum accurately is necessary because it feeds directly into the de Broglie wavelength equation.

Planck's constant \( h \) is a fundamental value in quantum mechanics, linking a particle's wavelength with its momentum. In the de Broglie equation, a particle's wavelength is defined by: \[ \lambda = \frac{h}{p} \] where \( \lambda \) is the de Broglie wavelength and \( p \) is momentum.
For electron wavelength calculation, Planck's constant is typically \( 6.63 \times 10^{-34} \, \text{J} \, \text{s} \). By substituting the previously calculated momentum and this constant, we find the de Broglie wavelength of an electron. The final computed wavelength for our example is approximately \( 1.10 \times 10^{-11} \, \text{m} \). This wavelength represents the wave-like properties of particles, showcasing the intriguing dual nature of matter.