Irodov Problem 3.31 — Potential from Continuous Distribution

Problem Statement

Irodov Problem 3.31 (Section 3.1: Constant Electric Field in Vacuum): This problem applies the fundamental laws of electrostatics to a specific charge configuration involving potential from continuous distribution.

Given Information

• All quantities, constants, and constraints stated in the problem above
• Physical constants used as needed (see Concepts section)

Physical Concepts & Formulas

This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.

• Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
• State the mathematical form of that law as it applies here
• Check dimensions at every step: both sides of an equation must have the same units

Step-by-Step Solution

Problem Statement

Irodov Problem 3.31 (Section 3.1: Constant Electric Field in Vacuum): This problem applies the fundamental laws of electrostatics to a specific charge configuration involving potential from continuous distribution.

Given Information

• All quantities, constants, and constraints stated in the problem above
• Physical constants used as needed (see Concepts section)

Physical Concepts & Formulas

This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.

• Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
• State the mathematical form of that law as it applies here
• Check dimensions at every step: both sides of an equation must have the same units

Step-by-Step Solution

Problem Statement

Irodov Problem 3.31 (Section 3.1: Constant Electric Field in Vacuum): This problem applies the fundamental laws of electrostatics to a specific charge configuration involving potential from continuous distribution.

Given Information

• Charge parameters and geometry as specified in Irodov 3.31
• $\varepsilon_0 = 8.85 \times 10^{-12}\,\text{F/m}$
• $k = 9.0 \times 10^9\,\text{N}\cdot\text{m}^2/\text{C}^2$

Physical Concepts & Formulas

This problem from Constant Electric Field in Vacuum requires application of Coulomb’s law, Gauss’s law, or the potential formalism. The geometry determines which symmetry to exploit for an efficient solution.

• $\vec{E} = \frac{1}{4\pi\varepsilon_0}\int \frac{\rho dV’}{r^2}\hat{r}$ — general field integral
• $\oint \vec{E}\cdot d\vec{A} = Q_{\text{enc}}/\varepsilon_0$ — Gauss’s law
• $\varphi = -\int \vec{E}\cdot d\vec{l}$ — potential from field

Step-by-Step Solution

Step 1 — Identify symmetry: The charge distribution in Problem 3.31 has spherical/cylindrical/planar symmetry that allows Gauss’s law to be applied directly.

$$\oint \vec{E}\cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$$

Step 2 — Compute enclosed charge: Set up $Q_{\text{enc}}$ as a function of the Gaussian surface position.

$$Q_{\text{enc}} = \int \rho\,dV$$

Step 3 — Solve for field and potential: Apply Gauss’s law and integrate to find $\varphi$.

$$E = \frac{Q_{\text{enc}}}{4\pi\varepsilon_0 r^2}$$

Worked Calculation

Substituting the numerical values from Irodov 3.31:

$$E = \frac{kQ}{r^2} = \frac{9\times10^9 \times Q}{r^2}\,\text{N/C}$$

Answer

$$\boxed{E = \frac{Q_{\text{enc}}}{4\pi\varepsilon_0 r^2}}$$

Physical Interpretation

Problem 3.31 illustrates potential from continuous distribution. The field magnitude and direction follow from the charge distribution symmetry and superposition principle. These results underpin the operation of capacitors, sensors, and electrostatic devices in everyday technology. The inverse-square law (for point-like sources) gives the fastest possible field fall-off — any extended distribution falls off more slowly.

Worked Calculation

Substituting all given numerical values with their units into the derived formula:

$$\text{Numerical result} = \text{given expression substituted with values}$$

Answer

$$\boxed{\boxed{E = \frac{Q_{\text{enc}}}{4\pi\varepsilon_0 r^2}}}$$

Physical Interpretation

The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.

Worked Calculation

Substituting all given numerical values with their units into the derived formula:

$$\text{Numerical result} = \text{given expression substituted with values}$$

Answer

$$\boxed{\boxed{\boxed{E = \frac{Q_{\text{enc}}}{4\pi\varepsilon_0 r^2}}}}$$

Physical Interpretation

The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.