Problem 18 - University Physics with Modern Physics

Question

Two stationary point charges \(+3.00 \mathrm{nC}\) and \(+2.00 \mathrm{nC}\) are separated by a distance of 50.0 \(\mathrm{cm}\) . An electron is released from rest at a point midway between the two charges and moves along the line connecting the two charges. What is the speed of the electron when it is 10.0 \(\mathrm{cm}\) from the \(+3.00-\mathrm{charge}\) ?

Step-by-Step Solution

Answer

The electron's speed is approximately 5.93 × 10^5 m/s.

Step 1: Identify Known Values and Constants

We have two charges, \(+3.00\,\mathrm{nC}\) and \(+2.00\,\mathrm{nC}\), separated by \(50.0\,\mathrm{cm}\). The electron is initially midway, and we want to find its speed \(v\) when it is \(10.0\,\mathrm{cm}\) away from the \(+3.00\,\mathrm{nC}\) charge. The charge of an electron, \(e\), is \(-1.60 \times 10^{-19}\,\mathrm{C}\), and its mass, \(m\), is \(9.11 \times 10^{-31}\,\mathrm{kg}\).

Step 2: Calculate Initial Electric Potential Energy

Initially, the electron is at the midpoint. The electric potential \(V\) at the midpoint due to the charges is the sum of the potentials due to each charge:\[ V = k\left(\frac{q_1}{r_1} + \frac{q_2}{r_2}\right) \]where \(k = 8.99 \times 10^9\,\mathrm{N\,m^2\,/\,C^2}\), \(r_1 = r_2 = 0.25\,\mathrm{m}\), \(q_1 = 3.00 \times 10^{-9}\,\mathrm{C}\), and \(q_2 = 2.00 \times 10^{-9}\,\mathrm{C}\). This gives:\[ V = 8.99 \times 10^9 \left(\frac{3.00 \times 10^{-9}}{0.25} + \frac{2.00 \times 10^{-9}}{0.25}\right) \]

Step 3: Compute Electric Potential Energy at Initial Point

The initial electric potential energy \(U_i\) is:\[ U_i = e \times V \]where \(e = 1.60 \times 10^{-19}\,\mathrm{C}\) (magnitude). Substitute the potential found in Step 2 to find \(U_i\).

Step 4: Calculate Electric Potential at the New Position

Now, the electron is \(10.0\,\mathrm{cm}\) (or \(0.10\,\mathrm{m}\)) from the \(+3.00\,\mathrm{nC}\) charge, making it \(50.0 - 10 = 40.0\,\mathrm{cm}\) (or \(0.40\,\mathrm{m}\)) from the \(+2.00\,\mathrm{nC}\) charge. Calculate the potential \(V_f\) again:\[ V_f = k\left(\frac{3.00 \times 10^{-9}}{0.10} + \frac{2.00 \times 10^{-9}}{0.40}\right) \]

Step 5: Compute Final Electric Potential Energy

The final electric potential energy \(U_f\) is:\[ U_f = e \times V_f \]Substitute the new potential from Step 4 here to find \(U_f\).

Step 6: Apply Conservation of Energy

According to the conservation of energy:\[ U_i + K_i = U_f + K_f \] Initially, \(K_i = 0\), so:\[ U_i = U_f + \frac{1}{2}mv^2 \]Rearrange to solve for \(v\):\[ v = \sqrt{\frac{2(U_i - U_f)}{m}} \]

Step 7: Calculate the Speed of the Electron

Using the values of \(U_i\) and \(U_f\) computed earlier, calculate the speed \(v\) of the electron. Substitute these values into the equation derived in Step 6 to obtain the final result for \(v\).

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle of electrostatics, describing the force between two point charges. It states that the electric force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Mathematically, it is expressed as:\[F = k\frac{|q_1q_2|}{r^2}\]where \(F\) is the force between the charges, \(k\) is Coulomb's constant \((8.99 \times 10^9 \, \text{N m}^2/\text{C}^2)\), \(q_1\) and \(q_2\) are the magnitudes of the charges, and \(r\) is the separation distance.

• This law helps in calculating the effect of each charge on the electron in the problem.
• It explains the interaction between the point charges and the moving electron.

This principle also allows us to determine the potential energy at any point, contributing to solving questions that involve multiple forces. In scenarios like ours, where an electron is between two charges, calculating the net force involves examining individual forces resulting from different charges. The calculations depend heavily on distance, making accurate values crucial.

Conservation of Energy

The Conservation of Energy principle states that the total energy in a closed system remains constant over time. It can neither be created nor destroyed but only transformed from one form to another. For our scenario, we focus on converting electric potential energy into kinetic energy.

• The initial potential energy of the electron when it's midway is transformed into kinetic energy as the electron moves.
• Using conservation equations, we compare initial and final total energies to solve for variables like velocity.

At the electron's initial position, all energy is in the form of electric potential energy \(U_i\). As the electron moves towards the charge, potential energy decreases while kinetic energy \(K\) increases. Thus, when applying conservation: \[U_i + K_i = U_f + K_f\]Since the electron starts from rest, \(K_i = 0\). Therefore, the equation simplifies to find the speed of the electron when considering changes in potential energy from its initial to final position.

Kinematics of Particles

Kinematics deals with the motion of objects without analyzing the forces causing the movement. In the context of charged particles like electrons, kinematics allows us to track their trajectory and speed based on energy transformations.

• Once we have the energy changes, kinematics provides the mathematical framework to derive velocity.
• By interpreting energy transformations, the speed at various points can be deduced.

For this problem, once the difference in potential energy is known using conservation laws, we employ kinematics to determine velocity. The kinematic equation derived specifically from the energy principle is:\[v = \sqrt{\frac{2(U_i - U_f)}{m}}\]This equation directly relates energy differences to motion parameters. Kinematics thus serves as a bridge from abstract energy concepts to practical calculations of motion characteristics like speed.