electromagnetic induction

Please refer to the following web-page that gives equations for the magnetic field surrounding a bar magnet,

http://instruct.tri-c.edu/fgram/web/mdipole.htm

Equation (4) on the page gives the value of magnetic flux density B along the axis of the magnet at short distances. Equation (3) gives the value of B when the distance from the magnet is large. It is in the form,

B = K/x^3,
where K is a constant characteristic to the dimensions and pole strength of the magnet, as given in equation (3)

From Faraday's law of electromagentic induction, the induced emf E, is given by,
E = -d(flux)/dt, where (flux) is the magnetic flux linked with the solenoid
For a solenoid with N turns and of cross-sectional area A,
E = -d(NAB(/dt0 = -NA[dB/dt]

Using B = K/x^3 for a bar magnet at large distance
dB/dt = (dB/dx).(dx/dt) = [-3K/x^4](dx/dt) = -(3K/x^4).V
where V is the speed of the bar magnet approaching the solenoid

Therefore, E = -NA[-(3K/x^4).V] = 3NAKV/x^4

Hence, for any instant of time, the induced emf is proportional to V. But the induced emf would increase with the decreasing of x.

When the bar magnet is at close proximity to the solenoid, dB/dx of equation (4) shown on the web-page needs to be performed.

Induced emf is directly proportional to V.

When V increases which means the "CHANGE of time" decreases, so by Faraday's law, the induced emf is increased.