We study the three point charges shown in the figure. They are held at the corners of an equilateral triangle with ℓ = 0.9 m. What is the electric potential energy of the system of three point charges? Use for the three charges

q1 = +2Q,

q2 = −3Q,
and
q3 = +Q,
where
Q = 101 nC.
Hint: The solution is done in steps. Assume that you first bring one of the point charges from a very large (infinite) distance to its position, then repeat this procedure for the second and third point charges

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To find the electric potential energy of the system of three point charges, we can follow these steps:

Step 1: Calculate the potential energy of bringing q1 to its position.
- q1 = +2Q = +2 * 101 nC = +202 nC
- The formula for electric potential energy is U = k * (q1 * q2) / r, where k is the electrostatic constant (9 x 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
- Since q1 is brought from an infinite distance, the potential energy is zero.

Step 2: Calculate the potential energy of bringing q2 to its position.
- q2 = -3Q = -3 * 101 nC = -303 nC
- The distance between q2 and q3 is ℓ = 0.9 m, as they are held at the corners of an equilateral triangle.
- The potential energy is U = k * (q2 * q3) / r, where r = ℓ.
- Plugging in the values, U = (9 x 10^9 N m^2/C^2) * (-303 nC) * (+101 nC) / (0.9 m).

Step 3: Calculate the potential energy of bringing q3 to its position.
- q3 = +Q = +101 nC
- The distance between q1 and q3 (or q2 and q3) is also ℓ = 0.9 m.
- Using the same formula as Step 2, U = (9 x 10^9 N m^2/C^2) * (+202 nC) * (+101 nC) / (0.9 m).

Step 4: Add the potential energies from Steps 1, 2, and 3 to find the total electric potential energy of the system.
- U_total = U1 + U2 + U3.

Please note that in steps 2 and 3, the negative sign of q2 is considered for the potential energy calculation, as it represents a charge of opposite sign.

To find the electric potential energy of the system of three point charges, we need to calculate the work done in bringing each charge individually from infinity to its position.

Step 1: Bringing q1 = +2Q from infinity to its position
Since q1 is brought from infinity, it starts with zero potential energy. The work done to bring q1 is equal to the change in potential energy.

The potential energy between two point charges is given by the equation:

U = k * |q1 * q2| / r

where U is the potential energy, k is the Coulomb's constant (9 x 10^9 N⋅m^2/C^2), q1 and q2 are the charges, and r is the distance between them.

In this case, q2 = -3Q and the distance between q1 and q2 is ℓ = 0.9 m, as they are held at the corners of an equilateral triangle.

Therefore, the potential energy between q1 and q2 is:

U1-2 = k * |q1 * q2| / ℓ

Substituting the values:

U1-2 = (9 x 10^9 N⋅m^2/C^2) * |(+2Q) * (-3Q)| / 0.9 m

Next, we need to account for the fact that q1 is also interacting with q3.

Step 2: Bringing q3 = +Q from infinity to its position
Again, q3 is brought from infinity, so it starts with zero potential energy. The work done to bring q3 is equal to the change in potential energy.

The potential energy between q1 and q3 is given by the same equation as before:

U = k * |q1 * q3| / r

In this case, q3 = +Q and the distance between q1 and q3 is also ℓ.

Therefore, the potential energy between q1 and q3 is:

U1-3 = k * |q1 * q3| / ℓ

Substituting the values:

U1-3 = (9 x 10^9 N⋅m^2/C^2) * |(+2Q) * (+Q)| / 0.9 m

Now, we need to account for the interaction between q2 and q3.

Step 3: Bringing q2 = -3Q from infinity to its position
Similar to the previous steps, the potential energy between q2 and q3 is given by the same equation:

U = k * |q2 * q3| / r

In this case, q2 = -3Q and the distance between q2 and q3 is also ℓ.

Therefore, the potential energy between q2 and q3 is:

U2-3 = k * |q2 * q3| / ℓ

Substituting the values:

U2-3 = (9 x 10^9 N⋅m^2/C^2) * |(-3Q) * (+Q)| / 0.9 m

Finally, to find the total electric potential energy of the system, we add up the potential energy differences for each pair of charges:

Total potential energy = U1-2 + U1-3 + U2-3

Plugging in the values calculated in each step, you can find the final answer.