Generating current through increasing capacitance

[Dilemma]

Hello,

1. Homework Statement

Assuming that the dielectric is inserted at a constant rate, find the current I as the slab is inserted.
Express your answer in terms of any or all of the given variables V, K, r, d, Δt, and ε0, the permittivity of free space.

Homework Equations

[/B]
C = ε0 * A / d

The Attempt at a Solution

[/B]
Q = CV
dQ = dC ⋅ V

I = dQ/Δt

C = ∈ ⋅ A / d = ∈⋅(xr)K / d + ∈⋅(r-x)r / d
dC = ∈⋅(r)K / d - ∈⋅r / d = ∈r(K-1)/d
dQ = dC⋅V

Therefore answer should be ∈r(K-1)V/(dΔt)

However, the correct answer is ∈(r^2)(K-1)V/(dΔt). That means the multiplier "r" is missing. What is wrong with my solution.

Thanks in advance.

 

[TSny]

dC = ∈⋅(r)K / d - ∈⋅r / d = ∈r(K-1)/d

The left side is a differential quantity. So, the right side should also be a differential quantity.

 

[Dilemma]

Oops, I have forgotten to include the dx multiplier. Thank you.

 

[TSny]

I = dQ/Δt

dQ is a differential quantity, but Δt is the finite amount of time required to completely insert the slab.
Did you mean to write I = dQ/dt, or maybe I = ΔQ/Δt?