Retarding Potential Required to Decelerate the Electron
To calculate the retarding potential required to decelerate the electron and bring it to a stop, we need to understand the concept of kinetic energy, potential energy, and conservation of energy.

Kinetic Energy of the Electron
The kinetic energy (KE) of an electron can be calculated using the formula:
KE = (1/2)mv^2

Given that the maximum velocity of the electron is 7 × 10^6 m/s, we can substitute this value into the formula:
KE = (1/2)m(7 × 10^6)^2

Potential Energy of the Electron
When the electron reaches the anode, all of its kinetic energy is converted into potential energy (PE). The potential energy is given by the formula:
PE = eV

Where e is the charge of the electron and V is the potential difference (retarding potential) across the anode.

Conservation of Energy
According to the law of conservation of energy, the total energy of a system remains constant. Therefore, the kinetic energy of the electron is equal to its potential energy when it reaches the anode. Mathematically, this can be expressed as:
KE = PE

Substituting the formulas for KE and PE, we get:
(1/2)m(7 × 10^6)^2 = eV

Calculating the Retarding Potential
To find the value of V, we need the mass of the electron (m) and the charge of the electron (e).

Mass of the Electron
The mass of the electron is approximately 9.11 × 10^-31 kg.

Charge of the Electron
The charge of the electron is equal to the elementary charge (e) and is approximately 1.6 × 10^-19 C.

Substituting the values into the equation:
(1/2)(9.11 × 10^-31 kg)(7 × 10^6 m/s)^2 = (1.6 × 10^-19 C)V

Simplifying the equation:
2.5355 × 10^-5 J = 1.6 × 10^-19 C × V

Solving for V:
V = (2.5355 × 10^-5 J) / (1.6 × 10^-19 C)

V ≈ 1.5847 × 10^14 V

Conclusion
To decelerate the electron and bring it to a stop, a retarding potential of approximately 1.5847 × 10^14 volts is required. This potential difference will convert the kinetic energy of the electron into potential energy as it reaches the anode.