Question

An electron is accelerated over a distance \(d\) by an electric potential V. The electric potential applied to this electron is then increased by a factor of 4 and the electron is accelerated over the same distance \(d\). The speed of the electron at the end of the second trial will be larger than at the end of the first trial by a factor of: (A) 16 (B) 8 (C) 4 (D) 2

Short answer

The speed of the electron at the end of the second trial will be larger by a factor of 2.

Step-by-step solution

Understand Kinetic Energy and Potential Difference

The kinetic energy (KE) gained by an electron accelerated through an electric potential V is given by the equation \[ KE = eV \].

Apply the Relationship to Both Trials

For the first trial, the kinetic energy is \[ KE_1 = eV \]. For the second trial (when potential is increased by a factor of 4), the kinetic energy is \[ KE_2 = e(4V) = 4eV \].

Relate Kinetic Energy to Velocity

Kinetic energy is also given by \[ KE = \frac{1}{2}mv^2 \]. Therefore, for the first trial,\[ \frac{1}{2}mv_1^2 = eV \] and for the second trial,\[ \frac{1}{2}mv_2^2 = 4eV \].

Solve for Velocities

From the first trial,\[ v_1 = \sqrt{\frac{2eV}{m}} \]and from the second trial,\[ v_2 = \sqrt{\frac{8eV}{m}} \].

Calculate the Ratio of Velocities

The ratio of velocities is given by \[ \frac{v_2}{v_1} = \frac{ \sqrt{\frac{8eV}{m}} }{ \sqrt{\frac{2eV}{m}} } = \sqrt{4} = 2 \].

Conclude the Factor Increase

The speed of the electron at the end of the second trial is larger by a factor of 2 compared to the first trial.

Key concepts

kinetic energy

Kinetic energy is the energy possessed by an object due to its motion. For an electron, it's essential to understand that when it is accelerated through an electric field, it gains kinetic energy. The relationship between the electron's kinetic energy (KE) and the electric potential (V) it experiences is given by \( KE = eV \),where 'e' is the elementary charge of the electron. This means if you increase the electric potential, the kinetic energy of the electron increases proportionally.

To illustrate:

• In the first trial of the problem, the kinetic energy is \( KE_1 = eV \) because the electron is accelerated by the potential difference V.
• In the second trial, with the potential difference increased by a factor of 4, the kinetic energy becomes \( KE_2 = 4eV \).

The increased kinetic energy means the electron moves faster in the second scenario.

electric potential

Electric potential is a measure of the potential energy per unit charge at a specific point in an electric field. When an electron is exposed to an electric potential, it experiences a force that accelerates it over a distance. This change in acceleration directly impacts the kinetic energy of the electron.

Here's how it works in two trials:

• In the first trial, the electron is subjected to an electric potential V over a distance d. The kinetic energy gained by the electron is \( KE_1 = eV \).
• In the second trial, the electric potential is increased to 4V (four times the original potential). Consequently, the kinetic energy becomes \( KE_2 = 4eV \).

This means that the electric potential directly influences the amount of kinetic energy the electron gains. By increasing the potential, the electron's speed and energy increase significantly.

velocity ratio

The velocity ratio between two scenarios is a comparison of an object's speed under different conditions. For the electron in this problem, the ratio compares its speed after being accelerated by two different electric potentials.
To find this ratio:

• First, use the kinetic energy expressions for both trials: \( KE_1 = \frac{1}{2}mv_1^2 = eV \) and \( KE_2 = \frac{1}{2}mv_2^2 = 4eV \).
• Simplify these to find the velocities: \( v_1 = \frac{2eV}{m} \) and \( v_2 = \frac{8eV}{m} \), which gives \( v_2 = 2v_1 \).

By dividing \( v_2 \) by \( v_1 \) (velocity at the second trial by the velocity at the first trial), we get the velocity ratio: \( \frac{v_2}{v_1} = 2 \). This means that the electron's speed is doubled in the second trial compared to the first.