Determine the electric potential energy for the array
of three charges in the drawing, relative to its value when the charges are
infinitely far away and infinitely far apart.
The drawing is a right triangle. The left side's length is 3.00m, the bottom side's length is 4.00m, and the hypotenuse's length is 5.00m. The top-left vertex is +8.00 microcoulombs, the bottom-left vertex is -15.0 microcoulombs, and the vertex on the right is +20.0 microcoulombs.
To determine the electric potential energy for the array of three charges, we can use the formula for electric potential energy U:
U = k * (q1 * q2) / r
where k is the electrostatic constant (8.99 * 10^9 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: +8.00 μC, -15.0 μC, and +20.0 μC.
Step 1: Calculate the electric potential energy between charges +8.00 μC and -15.0 μC.
q1 = +8.00 μC
q2 = -15.0 μC
r = distance between them (which is the length of the left side of the triangle, 3.00 m)
Convert the charges to coulombs:
q1 = +8.00 * 10^-6 C
q2 = -15.0 * 10^-6 C
Substitute the values into the formula:
U1 = (8.99 * 10^9 Nm^2/C^2) * [(+8.00 * 10^-6 C) * (-15.0 * 10^-6 C)] / 3.00 m
Step 2: Calculate the electric potential energy between charges +8.00 μC and +20.0 μC.
q1 = +8.00 μC
q2 = +20.0 μC
r = distance between them (which is the length of the bottom side of the triangle, 4.00 m)
Convert the charges to coulombs:
q1 = +8.00 * 10^-6 C
q2 = +20.0 * 10^-6 C
Substitute the values into the formula:
U2 = (8.99 * 10^9 Nm^2/C^2) * [(+8.00 * 10^-6 C) * (+20.0 * 10^-6 C)] / 4.00 m
Step 3: Calculate the electric potential energy between charges -15.0 μC and +20.0 μC.
q1 = -15.0 μC
q2 = +20.0 μC
r = distance between them (which is the length of the hypotenuse of the triangle, 5.00 m)
Convert the charges to coulombs:
q1 = -15.0 * 10^-6 C
q2 = +20.0 * 10^-6 C
Substitute the values into the formula:
U3 = (8.99 * 10^9 Nm^2/C^2) * [(-15.0 * 10^-6 C) * (+20.0 * 10^-6 C)] / 5.00 m
Step 4: Calculate the total electric potential energy for the array.
Add up the individual potential energies:
U = U1 + U2 + U3
Note: It's important to use proper units for all the values in the calculations.
Please plug in the values into the equations to calculate the electric potential energy for the array. Let me know if you need further assistance.
To determine the electric potential energy for the array of three charges in the drawing, we need to calculate the potential energy contribution from each pair of charges and then add them up.
The formula for the electric potential energy between two charges is given by:
PE = k * (q1 * q2) / r
Where:
- PE is the electric potential energy
- k is the Coulomb's constant (k = 9.0 x 10^9 N·m²/C²)
- q1 and q2 are the magnitudes of the charges
- r is the distance between the charges
Let's go step by step to calculate the potential energy contribution from each pair of charges:
First, the potential energy between the +8.00 microcoulombs charge and the -15.0 microcoulombs charge:
PE1 = (9.0 x 10^9 N·m²/C²) * (8.00 x 10^-6 C) * (-15.0 x 10^-6 C) / r1
Since the charges are located at the vertices of the triangle, and the hypotenuse acts as the distance between them, we can use the Pythagorean theorem to find the hypotenuse length (r1):
r1 = √(3.00m)^2 + (4.00m)^2
Next, the potential energy between the +8.00 microcoulombs charge and the +20.0 microcoulombs charge:
PE2 = (9.0 x 10^9 N·m²/C²) * (8.00 x 10^-6 C) * (20.0 x 10^-6 C) / r2
Again, we can use the Pythagorean theorem to find the distance (r2) between these charges:
r2 = √(3.00m)^2 + (5.00m)^2
Finally, the potential energy between the -15.0 microcoulombs charge and the +20.0 microcoulombs charge:
PE3 = (9.0 x 10^9 N·m²/C²) * (-15.0 x 10^-6 C) * (20.0 x 10^-6 C) / r3
Using the Pythagorean theorem, we find the distance (r3) between these charges:
r3 = (4.00m) + (5.00m)
Now, we can add up the potential energy contributions from each pair of charges:
Total PE = PE1 + PE2 + PE3
After calculating these individual potentials and adding them up, you will get the electric potential energy for the array of three charges relative to its value when the charges are infinitely far away and infinitely far apart.