Question

1. Treacherous Triangle Trickery. Consider a charge distribution consisting of an equilateral triangle with a point charge q fixed at each of its vertices. Let d be the distance between the center of the triangle and each vertex, let the triangle’s center be at the origin, and let one of its vertices lie on the x-axis at the point x = −d.

1.1. Compute the electric field at the center of the triangle by explicitly computing the sum of the electric fields due the charges at each vertex.

1.2. Let V (x,y) be the electric potential as a function of position. Compute an expression for V (x,y), and try to simplify it if possible.

1.3. If a point charge Q is placed at rest at the origin, will it remain at rest? Justify using electric potential and symbolic computation.

1.4. Sketch the graph of V (x, 0) versus x.

1.5. Compute the Taylor expansion of V (x, 0) about x = 0 up to the term of order x 2 .

1.6. If a point charge Q is placed at the origin and then given a suffciently small kick in the x-direction, will it remain in the vicinity for the origin forever? Does it depend on the sign of Q? Does it matter if the kick is to the left or right? Justify all answers carefully.

1.7. If there is a case where the charge Q will oscillates under a small push in the x-direction, determine the period of small oscillations if the charge in the center has mass m. If there is not such a case of oscillatory motion, explain how you know this.

Answer 1