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Wize University Physics Textbook (Master) > Electrostatic Forces and Electric Fields

Electric Fields of Special Charge Distributions

Properties of ConductorsPrevious Section
Gauss’s Law - IntroductionNext Section
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Electric Fields of Special Charge Distributions


There are certain charge configurations that appear more often than others, both for exam settings and real-life applications, that are worth mentioning.
  • Conducting charged sphere: This charge distribution has the same formula as the electric field for a point charge, but the electric field is zero inside the conductor.

∣E⃗(r)∣={0r<RQ4πr2ϵor>R\boxed{|\vec E(r)|=\begin{cases} 0 & r < R \\ \frac{Q}{4\pi r^2\epsilon_o} & r > R \end{cases} }
  • Conducting charged spherical shell:
  • For the space between the shells, the electric field depends only on the interior charge and uses the same formula as above.
  • For the space outside the shells, the electric field depends on the total charge on all of the shells.
  • Inside the conductors, the field is zero.

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  • Infinite line of charge or infinite conducting charged cylinder: The formula for this electric field distribution is given below. For the cylindrical case, the field is zero inside the conductor, but otherwise uses the same formula.

∣E⃗(r)∣=λ2πrϵo\boxed{|\vec E(r)|=\frac{\lambda }{2\pi r \epsilon_o}}

Wize Concept
The linear charge density λ\lambda is the amount of charge per unit length of the distribution (λ=Q/L\lambda=Q/L).
  • Infinite conducting cylindrical shell:
  • For the space between the shells, the formula used is the same as above, but only relies on the linear charge density of the interior line of charge (or cylinder).
  • For the space outside the shells, the formula depends on the total linear charge density on all of the shells.
  • Inside the conductors, the field is zero.

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  • Infinite thin plate: The electric field is constant for this geometry and depends only on the charge density of the plate.

∣E⃗∣=σ2ϵo\boxed{|\vec E|=\frac{\sigma}{2\epsilon_o}}

Wize Concept
The surface charge density σ\sigma is the amount of charge per unit surface area of the charge distribution (σ=Q/A\sigma=Q/A).
  • Infinite thick insulating plate: For this geometry, the electric field is constant outside of the plate, but it varies linearly inside the plate (equations shown below).

∣E⃗(z)∣={ρd2ϵo∣z∣<d/2ρzϵo∣z∣>d/2\boxed{|\vec E(z)|=\begin{cases} \frac{\rho d}{2\epsilon_o} & |z| < d/2 \\ \frac{\rho z}{\epsilon_o} & |z| > d/2 \end{cases} }

Wize Concept
The volume charge density ρ\rho is the amount of charge per unit volume of the charge distribution (ρ=Q/V\rho=Q/V).

Practice: Two Line Charges


Find the magnitude of the electric field in the figure below at any point in space. Assume the lines are infinite and each have a charge density of λ\lambda.
Hint: The magnitude of the electric field from a single line of charge is given as follows:
∣E⃗∣=λ2πrϵo|\vec E|=\frac{\lambda }{2\pi r \epsilon_o}

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Practice: Four Parallel Charged Plates


Consider four infinite parallel plates. The top plate and bottom plates each have surface charge density −σ-\sigmaand the middle two plates each have surface charge density +2σ+2\sigma .

Find the electric field in terms of σ\sigma and ϵo\epsilon_o in each region, and draw the electric field lines.

Hint: for one plate, the magnitude of the electric field is E=σ2εoE=\frac{\sigma}{2\varepsilon_o} .