Problem 4 - Master Resource Book in JEE Main Physics

Question

The binding energy of the innermost electron in tungsten is \(40 \mathrm{k} \mathrm{eV}\). To produce characteristic X-rays using a tungsten target in an X-rays tube, the potential difference between the cathode and anti cathode should be (a) \(V<40 \mathrm{kV}\) (b) \(V \leq 40 \mathrm{kV}\) (c) \(V>40 \mathrm{kV}\) (d) \(V=40 \mathrm{kV}\)

Step-by-Step Solution

Answer

(c) \( V > 40 \text{ kV} \).

Step 1: Understanding the Problem

We need to determine the minimum potential difference needed to produce X-rays from a tungsten target. The potential difference must provide enough energy to knock out the innermost electrons of tungsten having a binding energy of 40 keV.

Step 2: Conversion of Energy Forms

The binding energy of 40 keV implies that incoming electrons must have kinetic energy greater than or equal to this energy to ionize an electron. This kinetic energy is provided by the electric potential, calculated using the relation:\[\text{Kinetic Energy (eV)} = eV\]Where \( V \) is the potential difference.

Step 3: Setting up the Equation

For the innermost electron to be knocked out, the energy from the electric potential must at least equal the binding energy of the electron. Therefore,\[ eV \geq 40 \text{ keV} \]which simplifies to:\[ V \geq 40 \text{ kV} \]

Step 4: Analyzing Options

We compare the calculated inequality with the options provided:- Option (a) \( V < 40 \text{ kV} \) does not meet the required energy.- Option (b) \( V \leq 40 \text{ kV} \) could just match the energy, not ensuring the necessary overbinding.- Option (c) \( V > 40 \text{ kV} \) ensures sufficient energy beyond the binding energy.- Option (d) \( V = 40 \text{ kV} \) provides just enough energy but may not be practical due to energy losses.

Step 5: Conclusion

To produce characteristic X-rays by overcoming the binding energy, the most suitable choice is a potential difference greater than 40 kV.

Key Concepts

Binding Energy

Binding energy is a fundamental concept in physics. It refers to the energy required to remove an electron from its orbit around the nucleus of an atom. In an X-ray tube, where characteristic X-rays are produced, this concept is crucial.

• The electron in question, in our case, is the innermost or core electron of a tungsten atom.
• The binding energy plays a key role in determining if an incident electron can eject this core electron.

For tungsten, the binding energy is 40 keV. This means an external energy source must provide at least this amount of energy to knock out the electron. Once the core electron is ejected, the atom reaches an excited state and emits X-rays as it returns to a stable state. This emission is due to the vacancy created, which is filled by an electron from a higher energy level, thereby releasing energy as X-rays.

Potential Difference

The potential difference, often referred to as voltage, is another critical factor in the generation of X-rays. In an X-ray tube, it is the potential difference that accelerates electrons towards the target material.

• The electrons need sufficient kinetic energy to overcome the binding energy of the inner electrons of the target material.
• The kinetic energy of these electrons is directly proportional to the applied potential difference.

To ionize the innermost electron in the tungsten target, the potential difference must be greater than 40 kV. The equation used is: \[ Kinetic\ Energy\ (eV) = eV \]where \( e \) is the charge of the electron, and \( V \) is the potential difference. Therefore, to ensure the removal of an electron and the production of characteristic X-rays, \( V \) should be more than 40 kV.

Tungsten Target

The tungsten target is an essential component in the creation of characteristic X-rays. Tungsten is chosen primarily due to its high atomic number and its ability to withstand high temperatures.

• It is often used in X-ray tubes as the material where the accelerated electrons are made to collide.
• Upon collision, these high-energy electrons interact with the tungsten atoms, knocking out electrons and causing the emission of characteristic X-rays.

The high binding energy of tungsten's inner electrons allows for the production of high-energy X-rays, which are ideal for various applications, such as medical imaging and material analysis. In summary, the tungsten target facilitates the generation of X-rays by providing the high atomic number and thermal stability needed for efficient X-ray production. This choice of material not only ensures effective electron collision but also enhances the quality and intensity of the generated X-rays.