Consider the final arrangement of charged particles shown in the figure below. What is the work necessary to build such an arrangement of particles, assuming they were originally very far from one another? (Let q1 = 6.5 nC,q2 = 3.0 nC and q3 = −17.0 nC.) q2 = 4.5 nC and q3 = −18.5 nC.)
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To calculate the work necessary to build such an arrangement of charged particles, we need to find the potential energy of the system.
The potential energy of a system of charged particles can be calculated using the formula:
U = k * ( q1 * q2 / r12 + q1 * q3 / r13 + q2 * q3 / r23)
where U is the potential energy, k is the Coulomb constant (k ≈ 8.99 x 10^9 Nm^2/C^2), q1, q2, and q3 are the charges of the particles, and r12, r13, and r23 are the distances between the particles.
Given:
q1 = 6.5 nC
q2 = 3.0 nC
q3 = -17.0 nC
r12 = 0.050 m (-ve sign indicates that they are in opposite direction)
r13 = 0.075 m
r23 = 0.030 m (-ve sign indicates that they are in opposite direction)
Using the given values, we can substitute them into the formula:
U = (8.99 × 10^9 Nm^2/C^2) * [(6.5 × 10^−9 C)(3.0 × 10^−9 C) / 0.050 m + (6.5 × 10^−9 C)(−17.0 × 10^−9 C) / 0.075 m + (3.0 × 10^−9 C)(−17.0 × 10^−9 C) / 0.030 m]
Now we can calculate the expression:
U = (8.99 × 10^9 Nm^2/C^2) * (0.650 / 0.050 - 1.105 / 0.075 - 0.867 / 0.030) J
Simplifying the expression:
U ≈ 2.156 × 10^9 J
Therefore, the work necessary to build such an arrangement of particles, assuming they were originally very far from one another, is approximately 2.156 × 10^9 J.
To calculate the work necessary to build the final arrangement of charged particles, we can use the equation:
Work = potential energy of the final arrangement - potential energy of the initial arrangement
To find the potential energy of a system of charged particles, we can use the equation:
Potential energy = (k * |q1 * q2|) / r12 + (k * |q1 * q3|) / r13 + (k * |q2 * q3|) / r23
where k is Coulomb's constant (8.99 × 10^9 N*m^2/C^2), q1, q2, and q3 are the charges of the particles, and r12, r13, and r23 are the distances between the particles.
Let's calculate the potential energy of the final arrangement first:
1. Calculate the distances between the particles:
- r12 = the distance between q1 and q2
- r13 = the distance between q1 and q3
- r23 = the distance between q2 and q3
2. Substitute the values of q1, q2, q3, and r12, r13, and r23 into the potential energy equation and calculate the potential energy.
Repeat the same steps to calculate the potential energy of the initial arrangement, assuming the particles are very far from one another (which means they are effectively at infinity).
Finally, subtract the potential energy of the initial arrangement from the potential energy of the final arrangement to find the work necessary to build the arrangement.