Q.Two fixed, identical conducting plates $\alpha$ and $\beta$, each of surface area $S$, are held parallel to each other. Plate $\alpha$ (on the left) carries charge $-Q$ and plate $\beta$ (in the middle) carries charge $q$, with $Q > q > 0$. A third identical plate $\gamma$, free to move, is placed parallel to them on the far side of $\beta$ (to the right), at a distance $d$ from $\beta$; initially $\gamma$ is uncharged. The plate $\gamma$ is released and slides toward $\beta$, striking it. The collision is elastic, and the contact time is long enough for charge to redistribute between $\beta$ and $\gamma$ while they touch.

1. $E=\dfrac{Q-q}{2\varepsilon_0 S}$.
2. $q_\beta=\dfrac{Q+q}{2},\ q_\gamma=\dfrac{q-Q}{2}$. …

Model

A large plate carrying charge $\pm\!\;$(magnitude) over area $S$ acts like a charged sheet producing a uniform field of magnitude $\dfrac{|\text{charge}|}{2\varepsilon_0 S}$ on each side, directed away from the plate if positive, toward it if negative. Take $+x$ to the right (from $\alpha$ toward $\gamma$).

(a) Field on $\gamma$ before collision ($\gamma$ uncharged)

At $\gamma$'s location (to the right of both $\alpha$ and $\beta$):

• Due to $\alpha\,(-Q)$: magnitude $\dfrac{Q}{2\varepsilon_0 S}$, pointing toward $\alpha$ (left).
• Due to $\beta\,(+q)$: magnitude $\dfrac{q}{2\varepsilon_0 S}$, pointing away from $\beta$ (right).

Net (taking left as the resultant direction since $Q>q$):

(b) Charges after the collision

While $\beta$ and $\gamma$ touch they form a single conductor of total charge $q + 0 = q$, sitting next to plate $\alpha$ (charge $-Q$). For two separated conductors, the two outer faces carry equal charge $\dfrac{(\text{total of both})}{2}=\dfrac{-Q+q}{2}$, and the facing surfaces carry $\pm\dfrac{Q+q}{2}$. Since the merged $\beta$–$\gamma$ conductor has no charge on its internal contact interface:

• $\beta$ (inner plate, facing $\alpha$) carries the facing-surface charge: $\;q_\beta = \dfrac{Q+q}{2}.$
• $\gamma$ (outer plate) carries the outer-surface charge: $\;q_\gamma = \dfrac{q-Q}{2}\;(<0).$

Check: $q_\beta + q_\gamma = \dfrac{Q+q}{2}+\dfrac{q-Q}{2} = q.$ ✓

(c) Speed of $\gamma$ after travelling $d$

After the collision $\gamma$ carries $q_\gamma=\dfrac{q-Q}{2}$. The field on $\gamma$ from the other two plates:

• Due to $\alpha\,(-Q)$: $\dfrac{Q}{2\varepsilon_0 S}$ toward $\alpha$ (left).
• Due to $\beta\,\big(\tfrac{Q+q}{2}\big)$: $\dfrac{Q+q}{4\varepsilon_0 S}$ away from $\beta$ (right). …

Method: Charged Conducting Plates — Sheet-Field Superposition + Conductor Contact

This technique applies to any problem with several large parallel charged conducting plates, where you must find the field at some point, the charge redistribution after two plates touch, or the resulting motion of a free plate.

Steps

Step 1: Model each plate as a charged sheet, not a point charge

A large conducting plate of charge $Q$ spread over area $S$ produces a uniform field on each side, independent of distance from the plate — this is what makes the problem tractable without doing an integral each time.

directed away from the plate if $Q>0$, toward the plate if $Q<0$.

Step 2: Superpose the sheet fields at the point of interest

Pick a consistent axis. At any point, add up (with sign, since each field is one-dimensional along that axis) the contribution from every plate, using each plate's own sign convention from Step 1. A charge-free plate contributes nothing until it acquires charge.

Step 3: When two plates come into contact, treat them as one conductor

Two touching plates merge into a single conductor: total charge is conserved, and it redistributes so that no field exists inside the merged conductor. In a stack of parallel plates, this means the charge splits between the merged conductor's two exposed faces so as to balance whatever external field the other (untouched) plates produce — the outer face and the face adjacent to a neighbouring charged plate generally end up with different amounts, found by requiring internal consistency (zero net field inside the conductor) together with total-charge conservation. Once the two split apart again, each plate keeps the charge that was sitting on the face it retains. …