Three point charges, q₁, q₂, and q₃, are located at the vertices of an equilateral triangle. If q₁ = q₂, what value must q₃ have so that the total potential energy of the arrangement is zero?

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Quizplus · Accepted Answer

For potential energy to be zero, the charges must be in the state of equilibrium. The sum of forces acting on charges must be zero. This only happens when the net force on each charge is zero. In the case of an equilateral triangle, if two charges are equal, the third charge must be opposite in sign and equal in magnitude to make the net force zero. In this case, q₁ and q₂ have equal magnitude, so the third charge q₃ must have opposite sign and equal magnitude to the other two charges. Therefore, q₃ = -q₁. The potential energy of the system is then calculated using the formula: U = (kq^2)/r where k is Coulomb's constant, q is the magnitude of the charge, and r is the distance between the charges. For each pair of charges, the energy is: U1 = kq₁^2/d U2 = kq₂^2/d U3 = k(-q₁)^2/d where d is the distance between the charges. The total potential energy of the system is the sum of these energies: Utotal = U1 + U2 + U3 = kq₁^2/d + kq₂^2/d + kq₁^2/d = kq₁^2/d + kq₂^2/d - kq₁^2/d = kq₂^2/d This expression is not zero, so the potential energy of the system cannot be zero for any value of q₃, except when it equals zero. Therefore, the best choice is C, −0.5 q₁, which is the only value that makes the net charge of the system zero, leading to zero potential energy of the three point charges.

Three Point Charges, Q1, Q2, and Q3, Are Located at the Vertices

Three Point Charges, Q₁, Q₂, and Q₃, Are Located at the Vertices