What is the electric potential energy of a system of three equal 6.40 nC charges placed on the x-axis. The separation of two of the charges is 2.10 cm and the third charge is placed in their middle.The potential energy is:
Question
What is the electric potential energy of a system of three equal 6.40 nC charges placed on the x-axis. The separation of two of the charges is 2.10 cm and the third charge is placed in their middle.The potential energy is:
Solution 1
The electric potential energy (U) of a system of charges is given by the formula:
U = k * q1 * q2 / r
where:
- k is Coulomb's constant (8.99 * 10^9 N m^2/C^2),
- q1 and q2 are the charges, and
- r is the distance between the charges.
In this case, we have three charges (q1, q2, and q3) all equal to 6.40 nC = 6.40 * 10^-9 C. The distance between q1 and q2 is 2.10 cm = 2.10 * 10^-2 m, and q3 is placed in the middle, so the distance between each pair of charges is half of this, or 1.05 * 10^-2 m.
The total potential energy of the system is the sum of the potential energy between each pair of charges:
U_total = U_12 + U_23 + U_13
where:
- U_12 is the potential energy between q1 and q2,
- U_23 is the potential energy between q2 and q3, and
- U_13 is the potential energy between q1 and q3.
Since q1, q2, and q3 are all equal, and the distance between each pair of charges is the same, we can simplify this to:
U_total = 3 * U_12
Substituting the values into the formula gives:
U_total = 3 * (8.99 * 10^9 N m^2/C^2 * (6.40 * 10^-9 C)^2 / (1.05 * 10^-2 m))
Solving this gives the total electric potential energy of the system.
Solution 2
The electric potential energy (U) of a system of charges is given by the formula:
U = k * q1 * q2 / r
where:
- k is Coulomb's constant (8.99 * 10^9 N*m^2/C^2)
- q1 and q2 are the charges
- r is the distance between the charges
Given that we have three equal charges (q1 = q2 = q3 = 6.40 nC = 6.40 * 10^-9 C) and the separation between each pair is 2.10 cm = 2.10 * 10^-2 m / 2 = 1.05 * 10^-2 m (since the third charge is in the middle), we can calculate the potential energy of the system as the sum of the potential energy of each pair of charges.
So, the total potential energy (U_total) is:
U_total = U_12 + U_23 + U_13
where:
- U_12 is the potential energy between charges 1 and 2
- U_23 is the potential energy between charges 2 and 3
- U_13 is the potential energy between charges 1 and 3
Since the charges and the distances are the same for each pair, we can calculate the potential energy for one pair and then multiply by 3.
U_12 = k * q1 * q2 / r = 8.99 * 10^9 N*m^2/C^2 * 6.40 * 10^-9 C * 6.40 * 10^-9 C / 1.05 * 10^-2 m = 3.46 * 10^-9 J
So, the total potential energy is:
U_total = 3 * U_12 = 3 * 3.46 * 10^-9 J = 10.38 * 10^-9 J = 10.38 nJ
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