Problem Statement
A Van de Graaff generator has a sphere of radius 1.0 m. Find (a) the maximum potential the sphere can be raised to (breakdown field of air = $3\times10^6$ N/C), and (b) the charge at that potential.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$V_{max} = E_{max}\cdot R = 3\times10^6 \times 1.0 = 3\times10^6 \text{ V} = 3 \text{ MV}$$
$$Q = \frac{V_{max}}{k/R} = \frac{V_{max}\cdot R}{k} = \frac{3\times10^6\times1.0}{9\times10^9} = \frac{1}{3}\times10^{-3} \approx 333\,\mu\text{C}$$
$$\boxed{V_{max} = 3 \text{ MV},\quad Q \approx 333\,\mu\text{C}}$$
Answer
$$\boxed{V_{max} = 3 \text{ MV},\quad Q \approx 333\,\mu\text{C}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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