understand that, for a point outside a spherical conductor, the charge on the sphere may be considered to be a point charge at its centre

Uniform Electric Fields

Learning Objective

Understand that for a point outside a spherical conductor, the charge on the sphere may be considered to be a point charge located at its centre.

Key Concepts

• Conductors in electrostatic equilibrium have all excess charge residing on their surface.
• The electric field inside a conductor is zero.
• Gauss’s law relates the electric flux through a closed surface to the charge enclosed.
• For a spherically symmetric charge distribution, the field outside the distribution is identical to that of a point charge at the centre.

Why a Spherical Conductor Behaves Like a Point Charge

Consider a solid conducting sphere of radius \$R\$ carrying a total charge \$Q\$. The charge distributes uniformly over the surface because the conductor is an equipotential.

1. Choose a Gaussian surface – a sphere of radius \$r\$ such that \$r>R\$ and centred on the conductor.
2. By symmetry, the electric field \$\mathbf{E}\$ at every point on this Gaussian surface is radial and has the same magnitude \$E(r)\$.
3. Gauss’s law states

   \$\oint{\text{surface}} \mathbf{E}\cdot d\mathbf{A}= \frac{Q{\text{enc}}}{\varepsilon_0}.\$

   The left‑hand side becomes \$E(r)\,4\pi r^{2}\$ because \$d\mathbf{A}\$ is radial.

4. Since the only charge enclosed is the total charge \$Q\$ on the conductor,

   \$\$E(r)\,4\pi r^{2}= \frac{Q}{\varepsilon_0} \quad\Rightarrow\quad

   E(r)=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^{2}}.\$\$

   This is exactly the field of a point charge \$Q\$ placed at the centre.

Therefore, for any external point (\$r>R\$), the spherical conductor can be replaced by an equivalent point charge at its centre without changing the electric field.

Implications for Uniform Electric Fields

When multiple conductors are far apart, each can be treated as a point charge for the purpose of calculating the resultant field in the region between them. If the conductors are arranged so that the superposition of their fields is approximately constant over a region, that region experiences a uniform electric field.

Summary Table

Common Misconceptions

• “Charge spreads throughout the volume of a conductor.” – In electrostatic equilibrium, excess charge resides only on the outer surface.
• “The field inside a charged sphere is the same as outside.” – Inside a conducting sphere the field is zero; only a non‑conducting uniformly charged sphere has a non‑zero internal field.
• “The point‑charge model works only for very small spheres.” – It works for any spherical conductor as long as the observation point is outside the sphere.

Worked Example

Problem: A conducting sphere of radius \$0.10\ \text{m}\$ carries a charge of \$+5.0\ \mu\text{C}\$. Calculate the electric field \$5.0\ \text{cm}\$ above the surface along the radial line.

Solution:

1. Distance from centre: \$r = R + 0.05\ \text{m} = 0.10\ \text{m} + 0.05\ \text{m} = 0.15\ \text{m}\$.
2. Use point‑charge formula:

   \$\$E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^{2}}

   = \frac{9\times10^{9}\ \text{N·m}^{2}\!\!/\!\text{C}^{2}\times5.0\times10^{-6}\ \text{C}}{(0.15\ \text{m})^{2}}.\$\$

3. Calculate:

   \$\$E = \frac{9\times10^{9}\times5.0\times10^{-6}}{0.0225}

   \approx 2.0\times10^{6}\ \text{N·C}^{-1}.\$\$

Suggested Diagram

Key Take‑away

For any point external to a spherical conductor, the entire charge can be treated as if it were concentrated at the centre. This simplification follows directly from Gauss’s law and the symmetry of the sphere, and it underpins the analysis of uniform electric fields generated by multiple conductors.