Four electrons are located at the corners of a square 10.0nm on a side, with an alpha particle at its midpoint. How much work is needed to move the alpha particle to the midpoint of one of the sides of the square?
calculate the potential energy in the center
4 k 2 Qe^2/half diagonal^2
then calculate it at the center of a side
2 k 2 Qe^2/ half side^2 + 2 k 2 Qe^2/diagonal^2
the difference in potential energy is the work done
To calculate the work needed to move the alpha particle to the midpoint of one of the sides of the square, we can use the equation for electric potential energy:
Electric potential energy (U) = (k * Q1 * Q2) / r
Where:
- k is Coulomb's constant (8.99 x 10^9 N.m^2/C^2)
- Q1 and Q2 are the charges of the particles
- r is the distance between the particles
In this case, we have an alpha particle and four electrons. The charge of an alpha particle is +2e (where e is the elementary charge, 1.6 x 10^-19 C), and the charge of an electron is -e. The distance between the alpha particle and each electron is 10.0 nm (or 10.0 x 10^-9 m).
Let's calculate the electric potential energy for each pair of particles and then sum them up:
1. Alpha particle (charge: +2e) and electron 1 (charge: -e):
U1 = (k * (+2e) * (-e)) / (10.0 x 10^-9 m)
2. Alpha particle (charge: +2e) and electron 2 (charge: -e):
U2 = (k * (+2e) * (-e)) / (10.0 x 10^-9 m)
3. Alpha particle (charge: +2e) and electron 3 (charge: -e):
U3 = (k * (+2e) * (-e)) / (10.0 x 10^-9 m)
4. Alpha particle (charge: +2e) and electron 4 (charge: -e):
U4 = (k * (+2e) * (-e)) / (10.0 x 10^-9 m)
Now, let's calculate the total electric potential energy by summing up all four individual potentials:
Total U = U1 + U2 + U3 + U4
After calculating the total electric potential energy, we can find the work required to move the alpha particle to the midpoint of one of the sides of the square. The work (W) is equal to the change in potential energy:
W = Total U - U_initial
Since the alpha particle is initially at the midpoint of the square, all the potential energies are the same and cancel each other out. Therefore, U_initial = 0.
Thus, the work needed to move the alpha particle to the midpoint of one of the sides of the square is W = Total U.
You can now substitute the values and calculate the work needed using the formulas provided.
To calculate the work needed to move the alpha particle to the midpoint of one side of the square, we need to calculate the electrostatic potential energy of the system before and after the movement.
The electrostatic potential energy (U) between two charged particles is given by the formula:
U = (k * q1 * q2)/r
where k is Coulomb's constant (8.99 x 10^9 N m²/C²), q1 and q2 are the charges of the particles, and r is the distance between them.
Step 1: Calculate the potential energy before the movement.
In the initial configuration, the alpha particle is located at the midpoint of the square (distance = half the side length = 5.0 nm) and is surrounded by four electrons at the corners of the square.
The potential energy (U1) of the alpha particle with respect to each electron can be calculated using the above formula. Since the alpha particle and electrons have equal magnitude and opposite charges, their potential energy will be the same:
U1 = (k * q * q)/r = (8.99 x 10^9 N m²/C²) * [(2e)(-1e)] / (5.0 x 10^-9 m)
Note: e represents the elementary charge (1.6 x 10^-19 C).
Step 2: Calculate the potential energy after the movement.
After the movement, the alpha particle is located at the midpoint of one of the sides of the square. The distance between the alpha particle and each electron is now:
r' = √[(5.0 nm)^2 + (5.0 nm)^2] = 7.1 nm
The potential energy (U2) of the alpha particle with respect to each electron in this new configuration can be calculated in the same way:
U2 = (k * q * q) / r'
Step 3: Calculate the total work done.
The work done (W) to move the alpha particle from one configuration to the other is given by the difference in potential energy:
W = U2 - U1
Simply substitute the values into the equation and perform the calculations to find the answer.