Example 15 If the electron accelerated by a potential difference of 10 Vhas de Broglie wavelength of 10^-10m then electron accelerated by a potential difference of 20 VFind a wavelength of (nm)

To solve this problem, we can use the de Broglie equation, which relates the wavelength of a particle to its momentum:<br /><br />λ = h / p<br /><br />where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle.<br /><br />Given information:<br />- The electron is accelerated by a potential difference of 10 V.<br />- The de Broglie wavelength of the electron is 10^-10 m.<br /><br />We need to find the wavelength of the electron when it is accelerated by a potential difference of 20 V.<br /><br />Step 1: Calculate the momentum of the electron when accelerated by a potential difference of 10 V.<br />Using the de Broglie equation, we can write:<br /><br />λ = h / p<br />p = h / λ<br /><br />Substituting the given values:<br />p = h / (10^-10 m)<br />p = 6.626 × 10^-34 J·s / (10^-10 m)<br />p = 6.626 × 10^-24 kg·m/s<br /><br />Step 2: Calculate the momentum of the electron when accelerated by a potential difference of 20 V.<br />The momentum of the electron is directly proportional to the square root of the potential difference. Therefore, when the potential difference is doubled, the momentum will also be doubled.<br /><br />p' = 2 × p<br />p' = 2 × (6.626 × 10^-24 kg·m/s)<br />p' = 1.325 × 10^-23 kg·m/s<br /><br />Step 3: Calculate the wavelength of the electron when accelerated by a potential difference of 20 V.<br />Using the de Broglie equation again:<br />λ' = h / p'<br />λ' = 6.626 × 10^-34 J·s / (1.325 × 10^-23 kg·m/s)<br />λ' = 5.00 × 10^-11 m<br /><br />Therefore, the wavelength of the electron when accelerated by a potential difference of 20 V is 5.00 × 10^-11 m or 0.05 nm.