What is the wavelength of an electron accelerated through a potential difference of 100 V? (mass of electron = 9.11 x 10^-31 kg)

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Q: What is the wavelength of an electron accelerated through a potential difference of 100 V? (mass of electron = 9.11 x 10^-31 kg)

Solution: The wavelength λ = h/p = h/sqrt(2meV) = (6.63 x 10^-34)/sqrt(2 * 9.11 x 10^-31 * 100) = 1.22 x 10^-10 m.

Steps: 12

Step 1: Understand that we need to find the wavelength of an electron after it has been accelerated through a potential difference of 100 V.
Step 2: Recall the formula for the wavelength (λ) of a particle: λ = h/p, where h is Planck's constant and p is the momentum.
Step 3: Recognize that momentum (p) can be expressed in terms of mass (m) and velocity (v): p = mv.
Step 4: When an electron is accelerated through a potential difference (V), it gains kinetic energy equal to the work done by the electric field: KE = eV, where e is the charge of the electron.
Step 5: The kinetic energy can also be expressed as KE = (1/2)mv^2. Set these two expressions for kinetic energy equal: eV = (1/2)mv^2.
Step 6: Rearrange this equation to solve for v (velocity): v = sqrt(2eV/m).
Step 7: Substitute the values: e (charge of electron) = 1.6 x 10^-19 C, V = 100 V, and m (mass of electron) = 9.11 x 10^-31 kg into the equation to find v.
Step 8: Calculate v: v = sqrt(2 * (1.6 x 10^-19) * 100 / (9.11 x 10^-31)).
Step 9: Now, substitute v back into the momentum formula: p = mv = m * sqrt(2eV/m).
Step 10: Substitute this expression for p into the wavelength formula: λ = h/sqrt(2meV).
Step 11: Use Planck's constant h = 6.63 x 10^-34 J·s and substitute m = 9.11 x 10^-31 kg and V = 100 V into the equation: λ = (6.63 x 10^-34)/sqrt(2 * 9.11 x 10^-31 * 100).
Step 12: Calculate the final value to find the wavelength λ.