You are operating an X-ray tube with a cobalt (Co) target by applying an acceleration potential of 53 kVolt. Calculate the value of the characteristic wavelength λKα. Express your answer in meters.
To calculate the value of the characteristic wavelength λKα, we need to use the equation:
λ = hc / (E * (1 + E / (2mc^2)))
Where:
λ is the wavelength,
h is the Planck's constant (6.626 x 10^-34 J s),
c is the speed of light (3 x 10^8 m/s),
E is the acceleration potential (53 kV or 53,000 V),
m is the mass of the electron (9.11 x 10^-31 kg).
Firstly, we need to convert the acceleration potential from kilovolts to volts:
53 kV = 53,000 V
Now, we can substitute the values into the equation:
λ = (6.626 x 10^-34 J s) * (3 x 10^8 m/s) / (53,000 V * (1 + 53,000 V / (2 * (9.11 x 10^-31 kg) * (3 x 10^8 m/s)^2)))
Simplifying the equation further:
λ = (6.626 x 3) / (53,000 * (1 + (53,000 / (2 * (9.11 x 10^-31) * (3 x 10^8)^2))))
λ = (19.878 x 10^-34) / (53,000 + (53,000^2 / (2 * (9.11 x 10^-31) * (3 x 10^8)^2)))
Calculating the denominator:
denominator = 53,000 + (53,000^2 / (2 * (9.11 x 10^-31) * (3 x 10^8)^2))
denominator = 53,000 + (53,000^2 / (2 * 9.11 x 3^2 x 10^-31 x 10^16))
denominator = 53,000 + (53,000^2 / (2 * 9.11 x 3^2 x 10^-15))
Now, substitute the value of the denominator back into the equation:
λ = (19.878 x 10^-34) / denominator
Calculating the value of the characteristic wavelength requires evaluating the denominator and performing the division using the provided values.