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title: "An isolated conducting sphere of radius \\(R\\) is fixed in space and held at an electric potential \\(V_0\\) relative to infinity. A projectile point particle carrying a positive charge \\(q_0\\) and mass \\(m\\) is launched from very far away directly toward the center of the sphere. What is the minimum initial kinetic energy \\(K_{\\min}\\) required for the particle to just reach the surface of the sphere, and how does \\(K_{\\min}\\) depend on the particle’s mass \\(m\\)?"
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url: "https://nerd-notes.com/ubq/118069/"
date_modified: "2026-08-04T08:04:57+00:00"
---

# An isolated conducting sphere of radius \(R\) is fixed in space and held at an electric potential \(V_0\) relative to infinity. A projectile point particle carrying a positive charge \(q_0\) and mass \(m\) is launched from very far away directly toward the center of the sphere. What is the minimum initial kinetic energy \(K_{\min}\) required for the particle to just reach the surface of the sphere, and how does \(K_{\min}\) depend on the particle’s mass \(m\)?

An isolated conducting sphere of radius \(R\) is fixed in space and held at an electric potential \(V_0\) relative to infinity. A projectile point particle carrying a positive charge \(q_0\) and mass \(m\) is launched from very far away directly toward the center of the sphere. What is the minimum initial kinetic energy \(K_{\min}\) required for the particle to just reach the surface of the sphere, and how does \(K_{\min}\) depend on the particle's mass \(m\)?

![A conducting sphere of radius \(R\) centered at the origin is labeled with surface potential \(V_0\). Far to the right of the sphere along a dashed horizontal axis, a small circular point particle is labeled with charge \(q_0\) and mass \(m\). A straight horizontal arrow points from the particle directly left toward the sphere, labeled with initial velocity \(v_0\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830697-p82W2F.jpg)

- **A.** \(K_{\min} = q_0 V_0\); independent of \(m\)
- **B.** \(K_{\min} = q_0 V_0\); increases as \(m\) increases
- **C.** \(K_{\min} = \dfrac{1}{2}q_0 V_0\); independent of \(m\)
- **D.** \(K_{\min} = \dfrac{1}{2}q_0 V_0\); increases as \(m\) increases

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118069/*