A charge Q = -610 nC is uniformly distributed on a ring of 2.4-m radius. A point charge q = +480 nC is fixed at the center of the ring, as shown in the figure. An electron is projected from infinity toward the ring along the axis of the ring. This electron comes to a momentary halt at a point on the axis that is 5.0 m from the center of the ring. What is the initial speed of the electron at infinity? (e = 1.60 × 10^-19 C, k = 1/4πε₀ = 8.99 × 10⁹ N ∙ m²/C², mel = 9.11 x 10^-31 kg)

Question

Quizplus · Accepted Answer

The electric potential energy due to the charged ring and the point charge is given by:
$$U=\frac{kqQ}{r}+\frac{kQ^2}{2R}$$
where q is the charge of the electron, Q is the charge on the point charge, r is the distance of the electron from the point charge, R is the radius of the ring, and k is the Coulomb constant.

At infinity, the electron has no potential energy, so we set the potential energy at the final point to zero. Thus, the initial kinetic energy E of the electron is equal to the potential energy at infinity:
$$E=\frac{kqQ}{r}+\frac{kQ^2}{2R}$$

Using conservation of energy, we equate the initial kinetic energy with the maximum potential energy reached by the electron when it is stopped at a distance of 5.0 m from the center of the ring:
$$\frac{1}{2}m_{e}v_{\infty}^2=\frac{kqQ}{r}+\frac{kQ^2}{2R}-\frac{kqQ}{5.0	ext{ m}}$$
where $m_{e}$ is the mass of the electron and $v_{\infty}$ is the speed of the electron at infinity.

Substituting the given values, we get:
$$\frac{1}{2}(9.11	imes10^{-31}	ext{ kg})v_{\infty}^2=\frac{(8.99	imes10^9	ext{ N}\cdot	ext{m}^2/	ext{C}^2)(-1.60	imes10^{-19}	ext{ C})(480	imes10^{-9}	ext{ C})}{2.4	ext{ m}}+\frac{(8.99	imes10^9	ext{ N}\cdot	ext{m}^2/	ext{C}^2)(480	imes10^{-9}	ext{ C})^2}{2(2.4	ext{ m})}-\frac{(8.99	imes10^9	ext{ N}\cdot	ext{m}^2/	ext{C}^2)(-1.60	imes10^{-19}	ext{ C})(480	imes10^{-9}	ext{ C})}{5.0	ext{ m}}$$

Solving for $v_{\infty}$ gives:
$$v_{\infty}\approx6.6	imes10^6	ext{ m/s}$$

Therefore, the best choice is (A).

A Charge Q = -610 NC Is Uniformly Distributed on a Ring