Force between charged plates

[bodensee9]

Homework Statement

Two charged plates, plate 1 and plate 2, are separated by a distance s. Plate 1 has charge density $$\sigma_{1}$$ and plate 2 has charge density $$\sigma_{2}.$$ Assume that the electric field to the left of plate 1 is $$E_{1},$$ in between the 2 plates is $$E_{2},$$ and to the right of plate 2 is $$E_{3}.$$ Find the force per unit area of this configuration (I guess assume that somehow these two plates were connected).

The Attempt at a Solution

Assume Gaussian units:
So would I do $$dF = \int Edq$$ and so assume that the plates have a width called say r, then for plate 1,
$$F = \int E\sigma_{1}dr$$ from 0 to r
and since by Gauss' Law, $$dE = 4\pi\sigma_{1}$$, so I have
$$F = \int \frac{EdE}{4\pi}$$ evaluated from $$E_{1}to E_{2}.$$
Then I have $$F = \frac{E_{2}^{2}-E_{1}^{2}}{8\pi}.$$ BUt since I know that
$$E_{2}-E_{1} = 4\pi\sigma_{1}$$ so this is $$\frac{(E_{1} + E_{2})\sigma_{1}}{2}?$$ And that would be the force. I would do the same thing for plate 2? Thanks.

 

[anathema]

Your approach is correct. To find the force per unit area, we can use the definition of electric field as the force per unit charge. Therefore, the force per unit area on plate 1 would be:

F/A = (E2^2 - E1^2)/8π = (4πσ1)^2/8π = 2πσ1^2

Similarly, the force per unit area on plate 2 would be:

F/A = (E3^2 - E2^2)/8π = (4πσ2)^2/8π = 2πσ2^2

So, the total force per unit area on the plates would be the sum of these two values:

F/A = 2π(σ1^2 + σ2^2)

This is assuming that the plates are connected and can experience a net force due to the electric fields. I hope this helps. Let me know if you have any further questions or need clarification.

 

[tomcue]

I would first check the units of the equations provided to ensure they are consistent. It seems that the units are in Gaussian units, so I would convert them to SI units for ease of calculation and comparison to other equations.

Next, I would suggest using the equation F = qE, where F is the force, q is the charge, and E is the electric field. This equation can be applied to both plates separately, as they each have their own electric field (E1 and E3), and then the force per unit area can be calculated by dividing by the area of the plates.

For plate 1, the force per unit area would be F1/A = \sigma_{1}E_{1} where \sigma_{1} is the charge density and E1 is the electric field to the left of plate 1. Similarly, for plate 2, the force per unit area would be F2/A = \sigma_{2}E_{3} where \sigma_{2} is the charge density and E3 is the electric field to the right of plate 2.

To find the force per unit area between the plates, we can use the equation F2/A - F1/A = \sigma_{2}E_{3} - \sigma_{1}E_{1}. This represents the force per unit area created by the electric fields between the plates, since the electric fields are in opposite directions.

Finally, to find the total force per unit area of the configuration, we can add the force per unit area from each plate (F1/A and F2/A) to the force per unit area between the plates (\sigma_{2}E_{3} - \sigma_{1}E_{1}). This will give us the overall force per unit area.

In summary, the force per unit area between the charged plates can be found by using the equation F = qE and considering the electric fields and charge densities of each plate separately, as well as the electric field between the plates.