The problem of point charge between two conducting planes is...

The problem of point charge between two conducting planes is more easily tackled (if we want only the total charge induced on the planes) if we replace the point charge by a uniformly charged plane sheet. Let $\sigma$ be the charge density on this sheet and $E_{1}, E_{2}$ outward electric field on the two sides of this sheet. Then $\quad E_{1}+E_{2}=\frac{\sigma}{\varepsilon_{0}}$ The conducting planes will be assumed to be grounded. Then $E_{1} x=E_{2}(l-x)$. Hence $$ E_{1}=\frac{\sigma}{l \varepsilon_{0}}(l-x), E_{2}=\frac{\sigma}{l \varepsilon_{0}} x $$ This means that the induced charge density on the plane conductors are $$ \sigma_{1}=-\frac{\sigma}{l}(l-x), \sigma_{2}=-\frac{\sigma}{l} x $$ Hence $q_{1}=-\frac{q}{l}(l-x), q_{2}=-\frac{q}{l} x$

The problem of point charge between two conducting planes is more easily tackled (if we want only the total charge induced on the planes) if we replace the point charge by a uniformly charged plane sheet. Let $\sigma$ be the charge density on this sheet and $E_{1}, E_{2}$ outward electric field on the two sides of this sheet.
Then $\quad E_{1}+E_{2}=\frac{\sigma}{\varepsilon_{0}}$
The conducting planes will be assumed to be grounded. Then $E_{1} x=E_{2}(l-x)$.
Hence
$$
E_{1}=\frac{\sigma}{l \varepsilon_{0}}(l-x), E_{2}=\frac{\sigma}{l \varepsilon_{0}} x
$$
This means that the induced charge density on the plane conductors are
$$
\sigma_{1}=-\frac{\sigma}{l}(l-x), \sigma_{2}=-\frac{\sigma}{l} x
$$
Hence $q_{1}=-\frac{q}{l}(l-x), q_{2}=-\frac{q}{l} x$

Key Concepts

Gauss’s Law

Gauss’s Law relates the electric flux through a closed surface to the charge enclosed by that surface. In problems like this, it is used to connect the discontinuity in the electric field across a charged sheet to the surface charge density, forming a key relation for calculating induced charges.

Uniformly Charged Plane

A uniformly charged plane is an idealized model where a constant charge density is spread over an infinite or sufficiently large surface. This concept simplifies the analysis of electric fields and potentials by taking advantage of symmetry, which is especially useful in deriving relationships like the one between the electric field on either side of the sheet and the charge density.

Grounded Conductors

Grounded conductors are conductors connected to the earth, so their potential is fixed at zero. This boundary condition forces any excess charge to flow between the conductor and the ground until the potential is equalized. In the context of the problem, grounding the conducting planes ensures that the induced charge distribution is determined solely by the external charged sheet.

Induced Charge Distribution

Induced charge distribution refers to the redistribution of charges on the surface of a conductor in response to an external electric field or nearby charges. In this problem, the induced charges on the grounded planes adjust to cancel the field inside the conductors, and their total amounts can be determined by the relationships derived from the boundary conditions and Gauss’s Law.

Boundary Conditions in Electrostatics

Boundary conditions specify how electric fields and potentials behave at the interfaces between different materials. In electrostatic problems involving conductors, conditions such as the continuity of the tangential component of the electric field and the discontinuity in the normal component (proportional to the surface charge density) are critical. These conditions help derive the relationships used to understand how the electric fields created by the charged sheet interact with the grounded conducting planes.

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