Question

As an electron moves through a region of space, its speed decreases from $8.50 \times 10^{6} \mathrm{m} / \mathrm{s}\( to \)2.50 \times 10^{6} \mathrm{m} / \mathrm{s}$ The electric force is the only force acting on the electron. (a) Did the electron move to a higher potential or a lower potential? (b) Across what potential difference did the electron travel?

Short answer

Answer (b): The potential difference across which the electron traveled can be found by plugging the values calculated in the previous steps into the formula for \(\Delta V\).

Step-by-step solution

Understand the work-energy principle and electron motion in an electric field

The work-energy principle states that the work done on an object by external forces equals the change in its kinetic energy. For an electron moving in an electric field, the work done by the electric force causes a change in its kinetic energy which results in a change in potential energy and electric potential.

Calculate the change in kinetic energy

We need to find the change in kinetic energy for the electron. The kinetic energy of an electron can be calculated by the formula \(\frac{1}{2}mv^2\), where \(m\) is the mass of the electron and \(v\) is its speed. The change in kinetic energy is then given by \(\Delta KE = KE_f - KE_i\). Given that the initial speed of the electron is \(8.5\times 10^6\) m/s and the final speed is \(2.5\times 10^6\) m/s, we calculate the change in kinetic energy using the mass of an electron, \(m = 9.11\times 10^{-31}\) kg. \( \Delta KE = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2\)

Calculate the electric force's work

The work done by the electric force on the electron is equal to the change in its kinetic energy, which is negative since the electron's speed decreases. So, \(W = \Delta KE\).

Determine the direction of the electric force and potential

As the electric force is the only force acting on the electron, we can determine its direction by observing the change in speed. As the electron's speed decreases, the electric force must act opposite to the motion of the electron. The electric force on an electron is given by \(F = -eE\), where \(e\) is the electron's charge and \(E\) is the electric field. Since force and electric field are in the same direction, we can conclude that the electron moves towards a higher potential (opposite to electric field direction). Answer (a): The electron moves to a higher potential.

Calculate the potential difference

The relation between work done by electric force and potential difference is \(W = -e\Delta V\), where \(\Delta V\) is the potential difference. As the work done is equal to the change in kinetic energy, we have: \(-e\Delta V = \Delta KE\) Now, we can rearrange for \(\Delta V\): \(\Delta V=-\frac{\Delta KE}{e}\) The electron's charge is \(e=1.6\times 10^{-19}\) C. We can calculate the potential difference using the value of \(\Delta KE\) we obtained earlier: \(\Delta V=-\frac{\Delta KE}{1.6\times 10^{-19}}\) Finally, we compute the potential difference.

Final Answer

(a) The electron moves to a higher potential. (b) The potential difference across which the electron traveled can be determined by plugging the values calculated in the previous steps into the formula for \(\Delta V\).