In X - ray tube, accelerating voltage provides the energy to the electrons which produce X - rays. For getting X - rays, photons of 27.61 keV is required that the incident electrons must process kinetic energy at length 27.61 keV.

Given:
- The short wavelength end of the continuous spectrum of radiation produced by an X-ray tube is 0.45 Å.
- The order of accelerating voltage required in the tube is (10x) kV.

To find:
The value of x (integer).

Solution:
The continuous spectrum of radiation produced by an X-ray tube is due to the deceleration of high-speed electrons when they strike a metal target. The radiation emitted in this process is called bremsstrahlung radiation.

1. Relationship between wavelength and accelerating voltage:
The wavelength of the X-rays produced by an X-ray tube is inversely proportional to the accelerating voltage. Mathematically, this relationship can be represented as:

λ ∝ 1/V

Where:
- λ is the wavelength of the X-rays
- V is the accelerating voltage

2. Determining the relationship between wavelength and accelerating voltage:
We are given that the short wavelength end of the continuous spectrum is 0.45 Å. Let's denote this wavelength as λ₀.

Using the relationship mentioned above, we can write:

λ₀ ∝ 1/V

3. Converting wavelength to meters:
Since the given wavelength is in angstroms (Å), we need to convert it to meters to match the SI unit.

1 Å = 1 × 10^(-10) m

So, the short wavelength end (λ₀) can be written as:

λ₀ = 0.45 × 10^(-10) m

4. Solving for the accelerating voltage:
Substituting the value of λ₀ into the relationship between wavelength and accelerating voltage, we have:

0.45 × 10^(-10) m ∝ 1/V

Simplifying, we get:

V ∝ 1/(0.45 × 10^(-10) m)

5. Finding the value of x:
We are given that the order of accelerating voltage required is (10x) kV. Let's denote this value as V₀.

V₀ ∝ 1/(0.45 × 10^(-10) m)

To find x, we need to determine the exponent of 10 in V₀.

Writing V₀ in scientific notation, we have:

V₀ = 10^x kV

Substituting this into the equation above, we get:

10^x kV ∝ 1/(0.45 × 10^(-10) m)

Simplifying, we have:

10^x ∝ 1/(0.45 × 10^(-10) m)

Taking the logarithm of both sides, we get:

x log(10) ∝ log(1/(0.45 × 10^(-10) m))

Since log(10) = 1, we can simplify further:

x ∝ log(1/(0.45 × 10^(-10) m))

x ∝ log(1) - log(0.45 × 10^(-10) m)

x ∝ 0 - log(0.45 × 10^(-10) m)

x ∝ -log(0.45) - log(10^(-10) m)

Using logarithmic properties, we have:

x ∝ -log