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

Electric Fields of Special Charge Distributions

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.

Image illustrating Electric Fields of Special Charge Distributions

\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.

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.

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

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.

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

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

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).

\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} }

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 $\lambda$ .

Illustration for practice question: Two Line Charges

Hint: The magnitude of the electric field from a single line of charge is given as follows:

|\vec E|=\frac{\lambda }{2\pi r \epsilon_o}

$\frac{\lambda}{\pi \epsilon_o}\sqrt{\frac{1 }{x^2}+\frac{1 }{y^2}}$

$\frac{\lambda}{2\pi \epsilon_o}\sqrt{\frac{1 }{x^2}+\frac{1 }{y^2}}$

$\frac{\lambda}{2\pi \epsilon_o}\sqrt{\frac{1 }{x}+\frac{1 }{y}}$

$\frac{\lambda}{\pi \epsilon_o}\sqrt{\frac{1 }{x}+\frac{1 }{y}}$

$\frac{\lambda}{2\pi \epsilon_o}\sqrt{x^2+y^2}$

$\frac{\lambda}{\pi \epsilon_o}\sqrt{x^2+y^2}$

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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 $-\sigma$ and the middle two plates each have surface charge density $+2\sigma$ .

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

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

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Electric Fields of Special Charge Distributions

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