- #1

Terry Bing

- 48

- 6

## Homework Statement

(This is not a HW problem, but HW-type problem.)

A half cylinder of radius R and length L>>R is formed by cutting a cylindrical pipe made of an insulating material along a plane containing its axis. The rectangular base of the half cylinder is closed by a dielectric plate of length L and width 2R. A charge Q on the half cylinder and a charge q on the dielectric plate are uniformly sprinkled. Find the electrostatic force between the half cylinder and the dielectric plate.

## Homework Equations

Flux through a closed surface [itex]S[/itex] enclosing a charge [itex] q_{enc}[/itex] is

[tex]\oint_S \vec{E} \cdot \vec{dA} \ =\frac{q_{enc}}{\epsilon _0} [/tex]

Force [itex]\vec{dF} [/itex] on a charged surface carrying a surface charge density [itex]\sigma [/itex] due to an electric field [itex]\vec{E}[/itex] is

[tex]\vec{dF}=\sigma \vec{E} dA [/tex], where dA is the area element (scalar, not vector).

## The Attempt at a Solution

Brute force integration gave me

[tex]F=\frac{qQ}{8\epsilon_0 R L}[/tex]

Since this problem is from a high school textbook, I think brute force double integration [itex]\int_S \vec{dF}[/itex] is not the expected solution. So I tried the following:

For the flat plate which has uniform charge distribution,

[tex]\int_S \vec{dF}=\int_S\sigma \vec{E} dA [/tex]

By symmetry of the problem, we know that resultant force will be normal to the plate. Hence we pick the normal component of [itex]\vec{E}[/itex]. Let the magnitude of the net force be [itex]F[/itex]

[tex]\lvert \int_S \vec{dF} \rvert=\sigma \int_SE_\perp dA [/tex]

[tex]\implies F=\sigma \int_S \vec{E} \cdot \vec{dA} [/tex]

[tex]\implies F=\sigma \, \Phi [/tex]

where [itex] \Phi [/itex] is the flux due to the half cylinder through the plate.

[tex]\implies F=\frac{q}{2RL} \, \Phi [/tex]

Now, total flux emerging from the half cylinder is

[tex]\Phi_{tot} =\frac{Q}{\epsilon _0} [/tex]

Out of this if [itex]\Phi=\frac{1}{4}\Phi_{tot} =\frac{Q}{4 \epsilon _0} [/itex] passes through the plate, then we are done. But I don't see how I can show this.

Any help would be appreciated.

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