Two electrons are fixed 2.99 cm apart. Another electron is shot from infinity and stops midway between the two. What is its initial speed?
To find the initial speed of the electron, we can apply the principles of electrostatic forces and conservation of energy.
Step 1: Calculate the electric force between the two fixed electrons.
The electric force between two charges can be calculated using Coulomb's law:
F = k * (q1 * q2) / r^2,
where F is the electric force, k is Coulomb's constant (9 × 10^9 N m^2/C^2), q1 and q2 are the charges of the electrons, and r is the distance between them.
Since both electrons have the same charge (q1 = q2 = e, where e is the elementary charge), the formula simplifies to:
F = k * (e * e) / r^2.
Step 2: Calculate the work done by the electric field.
The electron travels from infinity to the midway point between the two fixed electrons. As it moves, the electric field does work on the electron, changing its kinetic energy. The work done by the electric field is equal to the change in kinetic energy of the electron.
Step 3: Apply conservation of energy.
The initial kinetic energy of the electron is zero when it is shot from infinity because its initial speed is zero. Therefore, the work done by the electric field equals the final kinetic energy of the electron:
W = ΔK.
Step 4: Solve for the initial speed.
The work done by the electric field can be calculated by multiplying the electric force by the distance traveled:
W = F * d,
where d is the distance from infinity to the midway point between the fixed electrons.
Since the work done equals the final kinetic energy of the electron, which is (1/2)mv^2, where m is the mass of the electron and v is its final velocity (speed), we can write:
F * d = (1/2)mv^2.
Substituting the expression for the electric force calculated in Step 1 and rearranging the equation, we have:
k * (e * e) / r^2 * d = (1/2)mv^2.
Solving for v:
v = √[2 * k * (e * e) * d / (r^2 * m)].
Now we can plug in the given values:
k = 9 × 10^9 N m^2/C^2, e = 1.6 × 10^-19 C (elementary charge), d = 2.99 cm = 0.0299 m, r = 0.0299 m (distance between the fixed electrons), and m = 9.11 × 10^-31 kg (mass of the electron).
Calculating v using the given values, we get:
v = √[2 * 9 × 10^9 * (1.6 × 10^-19)^2 * 0.0299 / (0.0299^2 * 9.11 × 10^-31)].