Problem Statement
A conducting sphere of radius $R$ is placed in a uniform external electric field $E_0$ (directed along the $z$-axis). Find the surface charge density $\sigma(\theta)$ induced on the sphere and verify that the total induced charge is zero.
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
A conducting sphere of radius $R$ is placed in a uniform external electric field $E_0$ (directed along the $z$-axis). Find the surface charge density $\sigma(\theta)$ induced on the sphere and verify that the total induced charge is zero.
Given Information
- $R$ = sphere radius
- $E_0$ = uniform external field magnitude
- $\theta$ = polar angle from $z$-axis
Physical Concepts & Formulas
A conducting sphere in an external field develops an induced surface charge that exactly cancels the external field inside. The solution uses the method of images or Legendre polynomial expansion. The induced charge distribution is a pure dipole layer (no monopole, no higher multipoles).
- $\sigma(\theta) = 3\varepsilon_0 E_0 \cos\theta$ — induced surface charge density
- $\sigma = \varepsilon_0 E_n$ — boundary condition at conductor surface
Step-by-Step Solution
Step 1 — Boundary conditions: Inside conductor: $\vec{E} = 0$. At surface: $E_{\text{tangential}} = 0$, and $E_{\text{normal}} = \sigma/\varepsilon_0$.
Step 2 — Potential outside: The total potential outside is the sum of external field potential ($-E_0 r\cos\theta$) plus the sphere’s induced dipole potential:
$$\varphi = -E_0 r\cos\theta + \frac{A\cos\theta}{r^2}$$
Step 3 — Apply boundary condition $\varphi = \text{const}$ at $r = R$:
$$-E_0 R\cos\theta + \frac{A\cos\theta}{R^2} = \text{const}$$
For this to be constant: $A = E_0 R^3$.
Step 4 — Compute normal field at surface:
$$E_r\bigg|_{r=R} = -\frac{\partial\varphi}{\partial r}\bigg|_{r=R} = E_0\cos\theta + \frac{2E_0 R^3}{R^3}\cos\theta = 3E_0\cos\theta$$
Step 5 — Surface charge density:
$$\sigma = \varepsilon_0 E_r|_{r=R} = 3\varepsilon_0 E_0\cos\theta$$
Step 6 — Total charge:
$$Q = \int_0^\pi \sigma(\theta) \cdot 2\pi R^2 \sin\theta\,d\theta = 3\varepsilon_0 E_0 \cdot 2\pi R^2 \int_0^\pi \cos\theta\sin\theta\,d\theta = 0 \checkmark$$
Worked Calculation
$$\sigma(\theta) = 3\varepsilon_0 E_0 \cos\theta$$
Maximum: $\sigma_{\max} = 3\varepsilon_0 E_0$ at $\theta = 0$ (facing field).
Zero: $\sigma = 0$ at $\theta = 90°$ (equator).
Minimum: $\sigma_{\min} = -3\varepsilon_0 E_0$ at $\theta = 180°$ (back).
Answer
$$\boxed{\sigma(\theta) = 3\varepsilon_0 E_0 \cos\theta}$$
Physical Interpretation
The induced charge is pure dipole — positive on the side facing the field, negative on the back. This is why conductors distort nearby electric field lines. The factor of 3 (versus 1 for a dielectric sphere) reflects the conductor’s perfect shielding. Lightning rods exploit this — the sharp tip concentrates $\sigma$, creating intense local fields that trigger lightning discharge.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{\sigma(\theta) = 3\varepsilon_0 E_0 \cos\theta}}$$
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.
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