Electric Potential Energy：練習問題

A thin non-conducting rod of length \(L\) carries a total charge \(Q\) distributed uniformly along its length. The rod lies along the x-axis with its center at the origin. What is the electric potential \(V\) at a point \(P\) on the x-axis at a distance \(d\) (where \(d > L/2\)) from the origin?

A very long coaxial cable consists of a thin inner conductor of radius \(a\) and a thin outer cylindrical shell of radius \(b\). The inner conductor carries a uniform linear charge density \(+\lambda\) and the outer shell carries \(-\lambda\). Find the energy stored per unit length in the electric field between the conductors.

An electric field is given by the expression \(\vec{E} = Ax^2 \hat{i}\), where \(A\) is a constant. Calculate the potential difference \(V(x_2) - V(x_1)\) between two points on the x-axis, where \(x_1 = L\) and \(x_2 = 2L\).

In a certain region of space, the electric potential is given by \(V(x, y, z) = 3x^2 y - y^3 z\). Find the y-component of the electric field, \(E_y\), at the point \((1, 1, 1)\).

Consider a semi-circular arc of radius \(R\) lying in the xy-plane with its center at the origin, extending from \(\theta = 0\) to \(\theta = \pi\). It carries a uniform linear charge density \(\lambda\). What is the electric potential at the origin?

A thin ring of radius \(R\) carries a total charge \(Q\) distributed uniformly along its circumference. Calculate the electric potential \(V\) at a point located on the central axis of the ring at a distance \(z\) from its center, and use this to find the work required to move a point charge \(q\) from infinity to this point.

Two point charges, \(+q\) and \(-2q\), are fixed on the x-axis at positions \(x = 0\) and \(x = d\) respectively.
(a) Find the location(s) on the x-axis where the electric potential \(V\) is zero (assuming \(V → 0\) at infinity).
(b) Calculate the total work required to bring a third charge \(+Q\) from infinity to the point midway between the two charges.

A thin insulating ring of radius \(R\) lies in the xy-plane and carries a total charge \(Q\) distributed uniformly along its circumference.
(a) Derive an expression for the electric potential \(V\) at a point \(P\) on the z-axis, located at a distance \(z\) from the center of the ring.
(b) Use the potential to find the electric field \(E\) at point \(P\).
(c) A small bead of mass \(m\) and charge \(-q\) is released from rest at \(z = z_0\) (where \(z_0 ≪ R\)). Describe the bead's motion and find its period of oscillation.