- #1

phantomvommand

- 241

- 37

- Homework Statement:
- The inner surfaces of 2 *insulating* plates are given a charge Q. What is the force required to hold them together?

- Relevant Equations:
- F = 1/2 (E outside + E inside) * Q

I have 2 methods, which give 2 different solutions:

Let sigma = charge per unit area

Let plate 1 be the left plate, plate 2 = right plate.

Method 1:

Because they are insulating, consider the electric field at 3 regions; region 1 to the left of plate 1, region 2 between the plates, and region 3 to the right of plate 2.

Because the plates are insulating, Electric field in the regions are superpositions of the field due to each plate.

Region 1 has field = sigma/e0 + 0 =

Region 2 has field = sigma/e0 - sigma/e0 =

Region 1 has field = sigma/e0 + 0 =

Average force on plate 1 = sigma/2e0 *Q (Average E field * Q)= Asigma^2/2e0 = Q^2/2Ae0, where A is area of plate. (force directed leftwards)

Similarly, force on plate 2 = Q^2/2Ae0. (force directed rightwards)

Thus, total force needed to hold the plates together is Q^2/Ae0.

Method 2:

Consider the capacitance of the system.

C = e0A/x, where x is plate separation.

Energy = Q^2x/2Ae0.

F = dE/dx = Q^2/2A e0.

Method 2 gives the correct answer. Why is method 1 wrong?

Let sigma = charge per unit area

Let plate 1 be the left plate, plate 2 = right plate.

Method 1:

Because they are insulating, consider the electric field at 3 regions; region 1 to the left of plate 1, region 2 between the plates, and region 3 to the right of plate 2.

Because the plates are insulating, Electric field in the regions are superpositions of the field due to each plate.

Region 1 has field = sigma/e0 + 0 =

**sigma/e0 directed leftwards**,Region 2 has field = sigma/e0 - sigma/e0 =

**0**,Region 1 has field = sigma/e0 + 0 =

**sigma/e0 directed rightwards**,Average force on plate 1 = sigma/2e0 *Q (Average E field * Q)= Asigma^2/2e0 = Q^2/2Ae0, where A is area of plate. (force directed leftwards)

Similarly, force on plate 2 = Q^2/2Ae0. (force directed rightwards)

Thus, total force needed to hold the plates together is Q^2/Ae0.

Method 2:

Consider the capacitance of the system.

C = e0A/x, where x is plate separation.

Energy = Q^2x/2Ae0.

F = dE/dx = Q^2/2A e0.

Method 2 gives the correct answer. Why is method 1 wrong?