How much work in eV is required to bring three charges of 5.4x10-19C from a great distance apart to 3.76x10-8m from one another (at the corners of an equilateral triangle)?
To calculate the work required to bring the charges together, we need to consider the electrostatic potential energy. The formula for potential energy is given by:
PE = k * (q1 * q2) / r
Where PE is the potential energy, k is the electrostatic constant (8.99x10^9 Nm^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between them.
In this case, we have three charges at the corners of an equilateral triangle. The distance between the charges is 3.76x10^-8 m. We need to calculate the potential energy for each pair of charges and add them up.
Let's break it down step by step:
1. Calculate the distance between two charges at the corners of an equilateral triangle:
The distance between adjacent charges in an equilateral triangle is given by the formula: r = a / √3, where a is the side length of the triangle.
In this case, we have r = 3.76x10^-8 m.
2. Calculate the potential energy for each pair of charges:
We'll have three pairs of charges, so we calculate the potential energy for each pair using the formula PE = k * (q1 * q2) / r.
The potential energy for each pair is:
PE1 = k * (q^2) / r
PE2 = k * (q^2) / r
PE3 = k * (q^2) / r
3. Calculate the total potential energy:
Total potential energy (PE_total) is the sum of the potential energies for each pair of charges.
PE_total = PE1 + PE2 + PE3
4. Convert the total potential energy to electron volts (eV):
1 electron volt (eV) is equal to 1.6x10^-19 J.
5. Calculate the work done:
The work done (W) is equal to the change in potential energy. Since the charges start from a great distance apart, the initial potential energy is zero.
Therefore, W = PE_total.
Now, let's plug in the values and calculate the work required.
Given:
q = 5.4x10^-19 C
r = 3.76x10^-8 m
k = 8.99x10^9 Nm^2/C^2
First, calculate the potential energy for each pair of charges:
PE1 = k * (q^2) / r
PE1 = (8.99x10^9 Nm^2/C^2) * ((5.4x10^-19 C)^2) / (3.76x10^-8 m)
PE1 = 8.62x10^-19 J
PE2 = k * (q^2) / r
PE2 = (8.99x10^9 Nm^2/C^2) * ((5.4x10^-19 C)^2) / (3.76x10^-8 m)
PE2 = 8.62x10^-19 J
PE3 = k * (q^2) / r
PE3 = (8.99x10^9 Nm^2/C^2) * ((5.4x10^-19 C)^2) / (3.76x10^-8 m)
PE3 = 8.62x10^-19 J
Next, calculate the total potential energy:
PE_total = PE1 + PE2 + PE3
PE_total = (8.62x10^-19 J) + (8.62x10^-19 J) + (8.62x10^-19 J)
PE_total = 2.58x10^-18 J
Now, convert the total potential energy to electron volts (eV):
1 eV = 1.6x10^-19 J
PE_total_eV = (2.58x10^-18 J) / (1.6x10^-19 J/eV)
PE_total_eV = 16.125 eV
Therefore, the work required to bring the charges together is approximately 16.125 eV.