prove that the charge induced does not depend on the rate of change of magnetic flux.

Explanation

Consider a rectangular wire loop PQRS of width `l`, with its plane perpendicular to a uniform magnetic field of induction `vecB`. The loop is being pulled out of the magnetic field at a constant speed v, as shown.
At any instant, let x be the length of the part of the loop in the magnetic field. Then, the magnetic flux through the loop is
`Phi=Blx " "` ...(1)

As the loop moves to the right through a distance dx = vdt in time dt, the area of the loop inside the field changes by `dA=ldx =lvdt. `
And, the change in the magnetic flux `d Phi` through the loop is
`d Phi =Bda=Blvdt " " ` ...(2)
Then, the time rate of change of magnetic flux is
`(d Phi)/(dt)=(Blvdt)/(dt)=Blv " " ` ...(3)
The changing magnetic flux induces a current `I` in the clockwise direction, as shown. a current-carrying conductor in a magnetic field experiences a force `vecF=I vecL xx vecB` (in the usual notation), whose direction can be found using Fleming's left hand rule. Accordingly, forces `vecF_(1) and vecF_(2)` on wires PS and QR, respectively, are equal in magnitude `(=IxB),` opposite in direction and have the same line of action. Hence, they balance each other. There is no force on wire RS as it lies outside the field.
The force `vecF_(3)` on wire PQ has magnitude `F_(3)=IlB` and is directed towards the left.
To move the loop with constant velocity `vec v`, an external force `vecF=-vecF_(3)` must be applied.
The work done by the external agent is
`dW=Fdx=-IlBdx=-IBdA=-Id Phi " " ` [from Eq. (2)] ...(4)
Therefore, the power, i.e., the time rate of doing work, is
`P=(dW)/(dt)=I(-(d Phi)/(dt)) " " ` ...(5)
The electric power when an emf E drives a current through a circuit is given by
`P=EI " " ` ...(6)
In Eq. (4), P is the electric power when a current `I` is driven through a circuit as a consequence of a change in the magnetic flux through it.
From Eqs. (5) and (6), `dW=Pdt =EIdt " " ` ...(7)
Therefore, comparing Eq. (4) with Eq. (7), `E=-(dPhi)/(dt) " " ` ...(8)
Thus, the emf induced in an electric circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit, which is Faraday-Lenz's law.