Four point charges of 89.67, 16.46, 12.1, and 19.76 microcoulombs are located at the corners of a rectangle of length 42 cm and width of 638 cm, as shown.
A sodium ion is placed at the point halfway between 16.46 and 12.1, and then released from rest. What will be its speed after it has traveled a distance of 45 km from where it was released? Assume it didn't bump into anything along the way, including any atoms or other charged particles, and ignore any effects that could be attributed to earth's gravity or magnetic field. In other words, keep the doggone problem simple! I've given you enough to think about just to be able to figure out the mass of a sodium ion!
To calculate the speed of the sodium ion after traveling a distance of 45 km, we need to follow a step-by-step process:
Step 1: Calculate the force experienced by the sodium ion due to the electric fields created by the four point charges.
Step 2: Calculate the work done by the electric fields on the sodium ion as it moves a distance of 45 km.
Step 3: Use the work-energy theorem to determine the change in kinetic energy of the sodium ion.
Step 4: Use the formula for kinetic energy to calculate the final speed of the sodium ion.
Let's start with Step 1:
1. Calculate the force experienced by the sodium ion due to the electric fields:
The electric force between two charges is given by Coulomb's Law:
F = k * (q1 * q2) / r^2
Where F is the electric force, k is Coulomb's constant (8.99 x 10^9 N m^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
We need to calculate the force due to each individual charge and then sum them up:
Let's denote the charges at the corners of the rectangle as:
Q1 = 89.67 μC
Q2 = 16.46 μC
Q3 = 12.1 μC
Q4 = 19.76 μC
Now, we can calculate the forces:
Forces due to Q1:
F1 = k * (q1 * q2) / r^2 = k * (Q1 * q2) / d^2
F1 = (8.99 x 10^9 N m^2/C^2) * (89.67 x 10^-6 C * q2) / (0.42 m)^2
Forces due to Q2:
F2 = k * (q2 * q3) / r^2 = k * (Q2 * q3) / w^2
F2 = (8.99 x 10^9 N m^2/C^2) * (16.46 x 10^-6 C * q3) / (0.638 m)^2
Forces due to Q3:
F3 = k * (q3 * q4) / r^2 = k * (Q3 * q4) / d^2
F3 = (8.99 x 10^9 N m^2/C^2) * (12.1 x 10^-6 C * q4) / (0.42 m)^2
Forces due to Q4:
F4 = k * (q4 * q1) / r^2 = k * (Q4 * q1) / w^2
F4 = (8.99 x 10^9 N m^2/C^2) * (19.76 x 10^-6 C * q1) / (0.638 m)^2
Step 2: Calculate the work done by the electric fields:
The work done by a force is given by the equation:
Work = Force * Distance
Since the sodium ion moves halfway between charges Q2 and Q3, the total distance it travels is given by:
Distance = (w / 2) + (d / 2) = (0.638 m / 2) + (0.42 m / 2)
Now, let's calculate the work done by each force:
Work1 = F1 * Distance
Work2 = F2 * Distance
Work3 = F3 * Distance
Work4 = F4 * Distance
Step 3: Use the work-energy theorem:
According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy:
Work_total = Change in Kinetic Energy
Since the sodium ion starts from rest, its initial kinetic energy is zero. Therefore, the work done by the electric fields will be equal to the change in kinetic energy:
Work_total = (1/2) * mass * (final velocity)^2 - (1/2) * mass * (initial velocity)^2
As the initial velocity is zero, the equation simplifies to:
Work_total = (1/2) * mass * (final velocity)^2
Step 4: Calculate the final speed of the sodium ion:
Using the work-energy theorem equation from Step 3, we can isolate (final velocity)^2:
(final velocity)^2 = (2 * Work_total) / mass
Now, we can substitute the obtained values of Work_total and mass into the equation and calculate the final velocity:
(final velocity)^2 = (2 * Work_total) / mass
(final velocity)^2 = (2 * (Work1 + Work2 + Work3 + Work4)) / mass
(final velocity)^2 = (2 * (Work1 + Work2 + Work3 + Work4)) / (mass of sodium ion)
Finally, take the square root of (final velocity)^2 to find the final speed of the sodium ion:
final velocity = √[(2 * (Work1 + Work2 + Work3 + Work4)) / (mass of sodium ion)]